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<div class="section" id="fwk-redden-ch05_s07" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">5.7</span> Complex Numbers and Their Operations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch05_s07_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch05_s07_o01" numeration="arabic">
<li>Define the imaginary unit and complex numbers.</li>
<li>Add and subtract complex numbers.</li>
<li>Multiply and divide complex numbers.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch05_s07_s01" version="5.0" lang="en">
<h2 class="title editable block">Introduction to Complex Numbers</h2>
<p class="para block" id="fwk-redden-ch05_s07_s01_p01">Up to this point the square root of a negative number has been left undefined. For example, we know that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1900" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow></math></span> is not a real number.</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1901" display="block"><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt><mo>=</mo><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext>or</mtext><mtext> </mtext><mtext> </mtext><msup><mrow><mrow><mo>(</mo><mrow><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mtext> </mtext><mo>−</mo><mn>9</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p03">There is no real number that when squared results in a negative number. We begin to resolve this issue by defining the <span class="margin_term"><a class="glossterm">imaginary unit</a><span class="glossdef">Defined as <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1902" display="inline"><mrow><mi>i</mi><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow></math></span> where <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1903" display="inline"><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow><mo>.</mo></math></span></span></span>, <em class="emphasis">i</em>, as the square root of −1.</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1904" display="block"><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>i</mi><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mtext> </mtext><mtext>and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s01_p05">To express a square root of a negative number in terms of the imaginary unit <em class="emphasis">i</em>, we use the following property where <em class="emphasis">a</em> represents any non-negative real number:</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1905" display="block"><mrow><msqrt><mrow><mo>−</mo><mi>a</mi></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mi>a</mi></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>⋅</mo><msqrt><mi>a</mi></msqrt><mo>=</mo><mi>i</mi><msqrt><mi>a</mi></msqrt></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s01_p07">With this we can write</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1906" display="block"><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>9</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>⋅</mo><msqrt><mn>9</mn></msqrt><mo>=</mo><mi>i</mi><mo>⋅</mo><mn>3</mn><mo>=</mo><mn>3</mn><mi>i</mi></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p09">If <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1907" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt><mo>=</mo><mn>3</mn><mi>i</mi></mrow></math></span>, then we would expect that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1908" display="inline"><mrow><mn>3</mn><mi>i</mi></mrow></math></span> squared will equal −9:</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1909" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mtext>−</mtext><mn>9</mn><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s01_p11">In this way any square root of a negative real number can be written in terms of the imaginary unit. Such a number is often called an <span class="margin_term"><a class="glossterm">imaginary number</a><span class="glossdef">A square root of any negative real number.</span></span>.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch05_s07_s01_p12">Rewrite in terms of the imaginary unit <em class="emphasis">i</em>.</p>
<ol class="orderedlist" id="fwk-redden-ch05_s07_s01_o01a" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1910" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>7</mn></mrow></msqrt></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1911" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>25</mn></mrow></msqrt></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1912" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>72</mn></mrow></msqrt></mrow></math></span></li>
</ol>
<p class="simpara">Solution:</p>
<ol class="orderedlist" id="fwk-redden-ch05_s07_s01_o01" numeration="loweralpha">
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1913" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>7</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>7</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>⋅</mo><msqrt><mn>7</mn></msqrt><mo>=</mo><mi>i</mi><msqrt><mn>7</mn></msqrt></mrow></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1914" display="inline"><mtable columnspacing="0.1em"><mtr><mtd style="padding-bottom:5%"><mrow><mtext> </mtext><msqrt><mrow><mtext>−</mtext><mn>25</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>25</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mn>25</mn></mrow></msqrt><mo>=</mo><mi>i</mi><mo>⋅</mo><mn>5</mn><mo>=</mo><mn>5</mn><mi>i</mi></mrow></mtd></mtr></mtable></math></span></li>
<li><span class="inlineequation"><math xml:id="fwk-redden-ch05_m1915" display="inline"><mrow><mtext> </mtext><msqrt><mrow><mtext>−</mtext><mn>72</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn><mo>⋅</mo><mn>36</mn><mo>⋅</mo><mn>2</mn></mrow></msqrt><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mn>36</mn></mrow></msqrt><mo>⋅</mo><msqrt><mn>2</mn></msqrt><mo>=</mo><mi>i</mi><mo>⋅</mo><mn>6</mn><mo>⋅</mo><msqrt><mn>2</mn></msqrt><mo>=</mo><mn>6</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow></math></span></li>
</ol>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s01_p13"><strong class="emphasis bold">Notation Note</strong>: When an imaginary number involves a radical, we place <em class="emphasis">i</em> in front of the radical. Consider the following:</p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1916" display="block"><mrow><mn>6</mn><mi>i</mi><msqrt><mn>2</mn></msqrt><mo>=</mo><mn>6</mn><msqrt><mn>2</mn></msqrt><mi>i</mi></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p15">Since multiplication is commutative, these numbers are equivalent. However, in the form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1917" display="inline"><mrow><mn>6</mn><msqrt><mn>2</mn></msqrt><mi>i</mi></mrow></math></span>, the imaginary unit <em class="emphasis">i</em> is often misinterpreted to be part of the radicand. To avoid this confusion, it is a best practice to place <em class="emphasis">i</em> in front of the radical and use <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1918" display="inline"><mrow><mn>6</mn><mi>i</mi><msqrt><mn>2</mn></msqrt><mo>.</mo></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p16">A <span class="margin_term"><a class="glossterm">complex number</a><span class="glossdef">A number of the form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1919" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>,</mo></mrow></math></span> where <em class="emphasis">a</em> and <em class="emphasis">b</em> are real numbers.</span></span> is any number of the form,
<span class="informalequation"><math xml:id="fwk-redden-ch05_m1920" display="block"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></math></span>
where <em class="emphasis">a</em> and <em class="emphasis">b</em> are real numbers. Here, <em class="emphasis">a</em> is called the <span class="margin_term"><a class="glossterm">real part</a><span class="glossdef">The real number <em class="emphasis">a</em> of a complex number <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1921" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></span></span> and <em class="emphasis">b</em> is called the <span class="margin_term"><a class="glossterm">imaginary part</a><span class="glossdef">The real number <em class="emphasis">b</em> of a complex number <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1922" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></span></span>. For example, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1923" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow></math></span> is a complex number with a real part of 3 and an imaginary part of −4. It is important to note that any real number is also a complex number. For example, 5 is a real number; it can be written as <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1924" display="inline"><mrow><mn>5</mn><mo>+</mo><mn>0</mn><mi>i</mi></mrow></math></span> with a real part of 5 and an imaginary part of 0. Hence, the set of real numbers, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1925" display="inline"><mi>ℝ</mi></math></span>, is a subset of the set of complex numbers, denoted <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1926" display="inline"><mi>ℂ</mi><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch05_s07_s01_p19"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1927" display="block"><mrow><mi>ℂ</mi><mo>=</mo><mrow><mo>{</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>|</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>ℝ</mi></mrow><mo>}</mo></mrow></mrow></math>
</span></p>
<div class="informalfigure large block">
<img src="section_08/de7ef4a536b8b16aa99d881cde47715f.png">
</div>
<p class="para block" id="fwk-redden-ch05_s07_s01_p21">Complex numbers are used in many fields including electronics, engineering, physics, and mathematics. In this textbook we will use them to better understand solutions to equations such as <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1928" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span> For this reason, we next explore algebraic operations with them.</p>
</div>
<div class="section" id="fwk-redden-ch05_s07_s02" version="5.0" lang="en">
<h2 class="title editable block">Adding and Subtracting Complex Numbers</h2>
<p class="para editable block" id="fwk-redden-ch05_s07_s02_p01">Adding or subtracting complex numbers is similar to adding and subtracting polynomials with like terms. We add or subtract the real parts and then the imaginary parts.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s02_n01">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch05_s07_s02_p02">Add: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1929" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s02_p03">Add the real parts and then add the imaginary parts.</p>
<p class="para" id="fwk-redden-ch05_s07_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1930" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi><mo>+</mo><mn>7</mn><mo>+</mo><mn>3</mn><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mo>+</mo><mn>7</mn><mo>−</mo><mn>2</mn><mi>i</mi><mo>+</mo><mn>3</mn><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>12</mn><mo>+</mo><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s02_p05">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1931" display="inline"><mrow><mn>12</mn><mo>+</mo><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s02_p06">To subtract complex numbers, we subtract the real parts and subtract the imaginary parts. This is consistent with the use of the distributive property.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s02_n02">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch05_s07_s02_p07">Subtract: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1932" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>10</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>9</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s02_p08">Distribute the negative sign and then combine like terms.</p>
<p class="para" id="fwk-redden-ch05_s07_s02_p09"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1933" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>10</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>9</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><mo>−</mo><mn>7</mn><mi>i</mi><mo>−</mo><mn>9</mn><mo>−</mo><mn>5</mn><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><mo>−</mo><mn>9</mn><mo>−</mo><mn>7</mn><mi>i</mi><mo>−</mo><mn>5</mn><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>1</mn><mo>−</mo><mn>12</mn><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s02_p10">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1934" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>12</mn><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s02_p11">In general, given real numbers <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em> and <em class="emphasis">d</em>:</p>
<p class="para block" id="fwk-redden-ch05_s07_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1935" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>c</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>+</mo><mi>d</mi></mrow><mo>)</mo></mrow><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>c</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>b</mi><mo>−</mo><mi>d</mi></mrow><mo>)</mo></mrow><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s07_s02_n03">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch05_s07_s02_p13">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1936" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1937" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn><mo>+</mo><mi>i</mi><mo>+</mo><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi><mo>−</mo><mn>4</mn><mo>+</mo><mn>7</mn><mi>i</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><mo>+</mo><mn>5</mn><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s02_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1938" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s02_p16">In summary, adding and subtracting complex numbers results in a complex number.</p>
</div>
<div class="section" id="fwk-redden-ch05_s07_s03" version="5.0" lang="en">
<h2 class="title editable block">Multiplying and Dividing Complex Numbers</h2>
<p class="para block" id="fwk-redden-ch05_s07_s03_p01">Multiplying complex numbers is similar to multiplying polynomials. The distributive property applies. In addition, we make use of the fact that <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1939" display="inline"><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow></math></span> to simplify the result into standard form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1940" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p02">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1941" display="inline"><mrow><mtext>−</mtext><mn>6</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p03">We begin by applying the distributive property.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p04"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1942" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>−</mtext><mn>6</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>6</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>⋅</mo></mrow></mstyle><mn>2</mn><mo>−</mo><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>6</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>⋅</mo></mrow></mstyle><mn>3</mn><mi>i</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>12</mn><mi>i</mi><mo>+</mo><mn>18</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>12</mn><mi>i</mi><mo>+</mo><mn>18</mn><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>12</mn><mi>i</mi><mo>−</mo><mn>18</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>18</mn><mo>−</mo><mn>12</mn><mi>i</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p05">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1943" display="inline"><mrow><mtext>−</mtext><mn>18</mn><mo>−</mo><mn>12</mn><mi>i</mi></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p06">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1944" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p07"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1945" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>3</mn><mo>⋅</mo></mrow></mstyle><mn>4</mn><mo>+</mo><mstyle color="#007fbf"><mrow><mn>3</mn><mo>⋅</mo></mrow></mstyle><mn>5</mn><mi>i</mi><mstyle color="#007fbf"><mrow><mo>−</mo><mn>4</mn><mi>i</mi><mo>⋅</mo></mrow></mstyle><mn>4</mn><mstyle color="#007fbf"><mrow><mo>−</mo><mn>4</mn><mi>i</mi><mo>⋅</mo></mrow></mstyle><mn>5</mn><mi>i</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn><mo>+</mo><mn>15</mn><mi>i</mi><mo>−</mo><mn>16</mn><mi>i</mi><mo>−</mo><mn>20</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>S</mi><mi>u</mi><mi>b</mi><mi>s</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn><mo>.</mo><mtext> </mtext></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn><mo>+</mo><mn>15</mn><mi>i</mi><mo>−</mo><mn>16</mn><mi>i</mi><mo>−</mo><mn>20</mn><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn><mo>−</mo><mi>i</mi><mo>+</mo><mn>20</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>32</mn><mo>−</mo><mi>i</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p08">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1946" display="inline"><mrow><mn>32</mn><mo>−</mo><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p09">In general, given real numbers <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em> and <em class="emphasis">d</em>:</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1947" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>d</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>d</mi><msup><mi>i</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>d</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>d</mi><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mi>c</mi><mo>+</mo><mrow><mo>(</mo><mrow><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi></mrow><mo>)</mo></mrow><mi>i</mi><mo>−</mo><mi>b</mi><mi>d</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>a</mi><mi>c</mi><mo>−</mo><mi>b</mi><mi>d</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi></mrow><mo>)</mo></mrow><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p11"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1948" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p12">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1949" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>12</mn><mi>i</mi></mrow></math></span></p>
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</div>
<p class="para block" id="fwk-redden-ch05_s07_s03_p14">Given a complex number <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1950" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></math></span>, its <span class="margin_term"><a class="glossterm">complex conjugate</a><span class="glossdef">Two complex numbers whose real parts are the same and imaginary parts are opposite. If given <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1951" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></math></span>, then its complex conjugate is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1952" display="inline"><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></span></span> is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1953" display="inline"><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span> We next explore the product of complex conjugates.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p15">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1954" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p16"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1955" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mn>5</mn><mo>⋅</mo></mrow></mstyle><mn>5</mn><mo>−</mo><mstyle color="#007fbf"><mrow><mn>5</mn><mo>⋅</mo></mrow></mstyle><mn>2</mn><mi>i</mi><mo>+</mo><mstyle color="#007fbf"><mrow><mn>2</mn><mi>i</mi><mo>⋅</mo></mrow></mstyle><mn>5</mn><mo>−</mo><mstyle color="#007fbf"><mrow><mn>2</mn><mi>i</mi><mo>⋅</mo></mrow></mstyle><mn>2</mn><mi>i</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn><mo>−</mo><mn>10</mn><mi>i</mi><mo>+</mo><mn>10</mn><mi>i</mi><mo>−</mo><mn>4</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn><mo>−</mo><mn>4</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>25</mn><mo>+</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>29</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p17">Answer: 29</p>
</div>
<p class="para block" id="fwk-redden-ch05_s07_s03_p18">In general, the <span class="margin_term"><a class="glossterm">product of complex conjugates</a><span class="glossdef">The real number that results from multiplying complex conjugates: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1956" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></span></span> follows:</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p19"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1957" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mo>⋅</mo><mi>b</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>⋅</mo><mi>a</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><msup><mi>i</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mi>i</mi><mo>+</mo><mi>a</mi><mi>b</mi><mi>i</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p20">Note that the result does not involve the imaginary unit; hence, it is real. This leads us to the very useful property</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p21"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1958" display="block"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p22">To divide complex numbers, we apply the technique used to rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator. The result can then be simplified into standard form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1959" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n04">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p23">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1960" display="inline"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p24">In this example, the conjugate of the denominator is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1961" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>.</mo></math></span> Therefore, we will multiply by 1 in the form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1962" display="inline"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p25"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1963" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>4</mn><mo>+</mo><mn>9</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>13</mn></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p26">To write this complex number in standard form, we make use of the fact that 13 is a common denominator.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p27"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1964" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>13</mn></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mrow><mn>13</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mi>i</mi></mrow><mrow><mn>13</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>2</mn><mrow><mn>13</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mn>13</mn></mrow></mfrac><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p28">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1965" display="inline"><mrow><mfrac><mn>2</mn><mrow><mn>13</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mn>13</mn></mrow></mfrac><mi>i</mi></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n05">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p29">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1966" display="inline"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mrow><mn>4</mn><mo>+</mo><mi>i</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p30"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1967" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mrow><mn>4</mn><mo>+</mo><mi>i</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>5</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo>+</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><mi>i</mi><mo>−</mo><mn>20</mn><mi>i</mi><mo>+</mo><mn>5</mn><msup><mi>i</mi><mn>2</mn></msup></mrow><mrow><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><mn>21</mn><mi>i</mi><mo>+</mo><mn>5</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>16</mn><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><mn>21</mn><mi>i</mi><mo>−</mo><mn>5</mn></mrow><mrow><mn>17</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>1</mn><mo>−</mo><mn>21</mn><mi>i</mi></mrow><mrow><mn>17</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>17</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>17</mn></mrow></mfrac><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p31">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1968" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>17</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>17</mn></mrow></mfrac><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p32">In general, given real numbers <em class="emphasis">a</em>, <em class="emphasis">b</em>, <em class="emphasis">c</em> and <em class="emphasis">d</em> where <em class="emphasis">c</em> and <em class="emphasis">d</em> are not both 0:</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p33"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1969" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>+</mo><mi>d</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>−</mo><mi>d</mi><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>c</mi><mo>−</mo><mi>d</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>a</mi><mi>c</mi><mo>−</mo><mi>a</mi><mi>d</mi><mi>i</mi><mo>+</mo><mi>b</mi><mi>c</mi><mi>i</mi><mo>−</mo><mi>b</mi><mi>d</mi><msup><mi>i</mi><mn>2</mn></msup></mrow><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo>−</mo><mi>a</mi><mi>d</mi><mo stretchy="false">)</mo><mi>i</mi></mrow><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mfrac><mrow><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi></mrow><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo><mo>+</mo><mo>(</mo><mfrac><mrow><mi>b</mi><mi>c</mi><mo>−</mo><mi>a</mi><mi>d</mi></mrow><mrow><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mrow></mfrac><mo>)</mo><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n06">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p34">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1970" display="inline"><mrow><mfrac><mrow><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p35">Here we can think of <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1971" display="inline"><mrow><mn>2</mn><mi>i</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></math></span> and thus we can see that its conjugate is <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1972" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mi>i</mi><mo>=</mo><mn>0</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p36"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1973" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mtext>−</mtext><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>16</mn><mi>i</mi><mo>+</mo><mn>6</mn><msup><mi>i</mi><mn>2</mn></msup></mrow><mrow><mtext>−</mtext><mn>4</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>16</mn><mi>i</mi><mo>+</mo><mn>6</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mtext>−</mtext><mn>4</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>16</mn><mi>i</mi><mo>−</mo><mn>6</mn></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>6</mn><mo>−</mo><mn>16</mn><mi>i</mi></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>−</mtext><mn>6</mn></mrow><mn>4</mn></mfrac><mo>−</mo><mfrac><mrow><mn>16</mn><mi>i</mi></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo><mn>4</mn><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p37">Because the denominator is a monomial, we could multiply numerator and denominator by 1 in the form of <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1974" display="inline"><mrow><mfrac><mi>i</mi><mi>i</mi></mfrac></mrow></math></span> and save some steps reducing in the end.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p38"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1975" display="block"><mrow><mtable columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>i</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>8</mn><mi>i</mi><mo>−</mo><mn>3</mn><msup><mi>i</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>8</mn><mi>i</mi><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo stretchy="false">(</mo><mtext>−</mtext><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>8</mn><mi>i</mi><mo>+</mo><mn>3</mn></mrow><mrow><mtext>−</mtext><mn>2</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>8</mn><mi>i</mi></mrow><mrow><mtext>−</mtext><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mtext>−</mtext><mn>2</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mtext>−</mtext><mn>4</mn><mi>i</mi><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p39">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1976" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo><mn>4</mn><mi>i</mi></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n06a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p40"><strong class="emphasis bold">Try this!</strong> Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1977" display="inline"><mrow><mfrac><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p41">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1978" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>i</mi></mrow></math></span></p>
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</div>
<p class="para block" id="fwk-redden-ch05_s07_s03_p43">When multiplying and dividing complex numbers we must take care to understand that the product and quotient rules for radicals require that both <em class="emphasis">a</em> and <em class="emphasis">b</em> are positive. In other words, if <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1979" display="inline"><mrow><mroot><mi>a</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1980" display="inline"><mrow><mroot><mi>b</mi><mpadded width="0.4em"><mi>n</mi></mpadded></mroot></mrow></math></span> are both real numbers then we have the following rules.</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p44"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1981" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#993300" mathvariant="bold"><mrow><mtext>Product rule for radicals</mtext></mrow></mstyle><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mroot><mrow><mi>a</mi><mo>⋅</mo><mi>b</mi></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>=</mo><mroot><mi>a</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>⋅</mo><mroot><mi>b</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mstyle color="#993300" mathvariant="bold"><mrow><mtext>Quotient rule for radicals</mtext></mrow></mstyle><mo>:</mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mroot><mrow><mfrac><mi>a</mi><mi>b</mi></mfrac></mrow><mpadded width="0.6em"><mi>n</mi></mpadded></mroot><mo>=</mo><mfrac><mrow><mroot><mi>a</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow><mrow><mroot><mi>b</mi><mpadded width="0.6em"><mi>n</mi></mpadded></mroot></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p45">For example, we can demonstrate that the product rule is true when <em class="emphasis">a</em> and <em class="emphasis">b</em> are both positive as follows:</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p46"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1982" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mn>4</mn></msqrt><mo>⋅</mo><msqrt><mn>9</mn></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>36</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mn>6</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mtext> </mtext><mstyle color="#007f3f"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p47">However, when <em class="emphasis">a</em> and <em class="emphasis">b</em> are both negative the property is not true.</p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p48"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1983" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow><mover><mo>=</mo><mrow><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow></mover></mrow></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>36</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>i</mi><mo>⋅</mo><mn>3</mn><mi>i</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>6</mn><msup><mi>i</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mtext>−</mtext><mn>6</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch05_s07_s03_p49">Here <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1984" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1985" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow></math></span> both are not real numbers and the product rule for radicals fails to produce a true statement. Therefore, to avoid some common errors associated with this technicality, ensure that any complex number is written in terms of the imaginary unit <em class="emphasis">i</em> before performing any operations.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n07">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p50">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1986" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mtext>−</mtext><mn>15</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p51">Begin by writing the radicals in terms of the imaginary unit <em class="emphasis">i</em>.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p52"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1987" display="block"><mrow><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mtext>−</mtext><mn>15</mn></mrow></msqrt><mo>=</mo><mi>i</mi><msqrt><mn>6</mn></msqrt><mo>⋅</mo><mi>i</mi><msqrt><mrow><mn>15</mn></mrow></msqrt></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p53">Now the radicands are both positive and the product rule for radicals applies.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p54"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1988" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mo>⋅</mo><msqrt><mrow><mtext>−</mtext><mn>15</mn></mrow></msqrt></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>i</mi><msqrt><mn>6</mn></msqrt><mo>⋅</mo><mi>i</mi><msqrt><mrow><mn>15</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>i</mi><mn>2</mn></msup><msqrt><mrow><mn>6</mn><mo>⋅</mo><mn>15</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><msqrt><mrow><mn>90</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><msqrt><mrow><mn>9</mn><mo>⋅</mo><mn>10</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>3</mn><mo>⋅</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><msqrt><mrow><mn>10</mn></mrow></msqrt></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p55">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1989" display="inline"><mrow><mtext>−</mtext><mn>3</mn><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n08">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p56">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1990" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>10</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p57">Begin by writing the radicals in terms of the imaginary unit <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1991" display="inline"><mi>i</mi></math></span> and then distribute.</p>
<p class="para" id="fwk-redden-ch05_s07_s03_p58"><span class="informalequation"><math xml:id="fwk-redden-ch05_m1992" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msqrt><mrow><mtext>−</mtext><mn>10</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>i</mi><msqrt><mrow><mn>10</mn></mrow></msqrt><mrow><mo>(</mo><mrow><mi>i</mi><msqrt><mn>6</mn></msqrt><mo>−</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>i</mi><mn>2</mn></msup><msqrt><mrow><mn>60</mn></mrow></msqrt><mo>−</mo><mi>i</mi><msqrt><mrow><mn>100</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><msqrt><mrow><mn>4</mn><mo>⋅</mo><mn>15</mn></mrow></msqrt><mo>−</mo><mi>i</mi><msqrt><mrow><mn>100</mn></mrow></msqrt></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>2</mn><mo>⋅</mo><msqrt><mrow><mn>15</mn></mrow></msqrt><mo>−</mo><mi>i</mi><mo>⋅</mo><mn>10</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>2</mn><msqrt><mrow><mn>15</mn></mrow></msqrt><mo>−</mo><mn>10</mn><mi>i</mi></mtd></mtr></mtable></math>
</span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p59">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1993" display="inline"><mrow><mtext>−</mtext><mn>2</mn><msqrt><mrow><mn>15</mn></mrow></msqrt><mo>−</mo><mn>10</mn><mi>i</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch05_s07_s03_p60">In summary, multiplying and dividing complex numbers results in a complex number.</p>
<div class="callout block" id="fwk-redden-ch05_s07_s03_n08a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch05_s07_s03_p61"><strong class="emphasis bold">Try this!</strong> Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1994" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mi>i</mi><msqrt><mn>5</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch05_s07_s03_p62">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1995" display="inline"><mrow><mtext>−</mtext><mn>12</mn><mo>+</mo><mn>6</mn><mi>i</mi><msqrt><mn>5</mn></msqrt></mrow></math></span></p>
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</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch05_s07_s03_n09">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch05_s07_s03_l01" mark="bullet">
<li>The imaginary unit <em class="emphasis">i</em> is defined to be the square root of negative one. In other words, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1996" display="inline"><mrow><mi>i</mi><mo>=</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1997" display="inline"><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow><mo>.</mo></math></span>
</li>
<li>Complex numbers have the form <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1998" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></math></span> where <em class="emphasis">a</em> and <em class="emphasis">b</em> are real numbers.</li>
<li>The set of real numbers is a subset of the complex numbers.</li>
<li>The result of adding, subtracting, multiplying, and dividing complex numbers is a complex number.</li>
<li>The product of complex conjugates, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m1999" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2000" display="inline"><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow></math></span>, is a real number. Use this fact to divide complex numbers. Multiply the numerator and denominator of a fraction by the complex conjugate of the denominator and then simplify.</li>
<li>Ensure that any complex number is written in terms of the imaginary unit <em class="emphasis">i</em> before performing any operations.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s07_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd01">
<h3 class="title">Part A: Introduction to Complex Numbers</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch05_s07_qs01_p01"><strong class="emphasis bold">Rewrite in terms of imaginary unit <em class="emphasis">i</em>.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2001" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>81</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2003" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>64</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2005" display="inline"><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2007" display="inline"><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>36</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2009" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>20</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2011" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>18</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2013" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>50</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2015" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>48</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2017" display="inline"><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>45</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2019" display="inline"><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>8</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2021" display="inline"><mrow><msqrt><mrow><mo>−</mo><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2023" display="inline"><mrow><msqrt><mrow><mo>−</mo><mfrac><mn>2</mn><mn>9</mn></mfrac></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2025" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>0.25</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2027" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>1.44</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd01_qd02" start="15">
<p class="para" id="fwk-redden-ch05_s07_qs01_p30"><strong class="emphasis bold">Write the complex number in standard form</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2029" display="inline"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>.</mo></math></span></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2030" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2032" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>5</mn><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2034" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mo>+</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>8</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2036" display="inline"><mrow><mn>4</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mtext>−</mtext><mn>18</mn></mrow></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2038" display="block"><mrow><mfrac><mrow><mn>3</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>24</mn></mrow></msqrt></mrow><mn>6</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2040" display="block"><mrow><mfrac><mrow><mn>2</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>75</mn></mrow></msqrt></mrow><mrow><mn>10</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa21">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2042" display="block"><mrow><mfrac><mrow><msqrt><mrow><mtext>−</mtext><mn>63</mn></mrow></msqrt><mo>−</mo><msqrt><mn>5</mn></msqrt></mrow><mrow><mtext>−</mtext><mn>12</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa22">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2044" display="block"><mrow><mfrac><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>72</mn></mrow></msqrt><mo>+</mo><msqrt><mn>8</mn></msqrt></mrow><mrow><mtext>−</mtext><mn>24</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd01_qd03" start="23">
<p class="para" id="fwk-redden-ch05_s07_qs01_p47"><strong class="emphasis bold">Given that</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2046" display="inline"><mrow><msup><mi>i</mi><mn>2</mn></msup><mo>=</mo><mtext>−</mtext><mn>1</mn></mrow></math></span> <strong class="emphasis bold">compute the following powers of</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2047" display="inline"><mi>i</mi><mo>.</mo></math></span></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2048" display="inline"><mrow><msup><mi>i</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2050" display="inline"><mrow><msup><mi>i</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2051" display="inline"><mrow><msup><mi>i</mi><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2053" display="inline"><mrow><msup><mi>i</mi><mn>6</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2054" display="inline"><mrow><msup><mi>i</mi><mrow><mn>15</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2056" display="inline"><mrow><msup><mi>i</mi><mrow><mn>24</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd02">
<h3 class="title">Part B: Adding and Subtracting Complex Numbers</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd02_qd01" start="29">
<p class="para" id="fwk-redden-ch05_s07_qs01_p60"><strong class="emphasis bold">Perform the operations.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2057" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>7</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2059" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2061" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2063" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>10</mn><mo>+</mo><mn>15</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>15</mn><mo>−</mo><mn>20</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2065" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2067" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>10</mn></mrow></mfrac><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2069" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>8</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2071" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>6</mn><mo>−</mo><mn>9</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2073" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>9</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>8</mn><mo>+</mo><mn>12</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2075" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>11</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>13</mn><mo>−</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2077" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>14</mn></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>7</mn></mfrac><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2079" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>8</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p85"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2081" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>+</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>6</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2083" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>6</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2085" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2087" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>−</mo><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2089" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mn>7</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>10</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2090" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>−</mo><mn>11</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>8</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2092" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>16</mn></mrow></msqrt><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2094" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>100</mn></mrow></msqrt><mo>+</mo><mrow><mo>(</mo><mrow><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2096" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2098" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>81</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2099" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><msqrt><mrow><mtext>−</mtext><mn>25</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>3</mn><mo>+</mo><mn>4</mn><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2101" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>12</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>49</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd03">
<h3 class="title">Part C: Multiplying and Dividing Complex Numbers</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd03_qd01" start="53">
<p class="para" id="fwk-redden-ch05_s07_qs01_p109"><strong class="emphasis bold">Perform the operations.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2103" display="inline"><mrow><mi>i</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p112"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2105" display="inline"><mrow><mi>i</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2107" display="inline"><mrow><mn>2</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>7</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2109" display="inline"><mrow><mn>6</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2111" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2113" display="inline"><mrow><mtext>−</mtext><mn>5</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2115" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2117" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2119" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>8</mn><mo>−</mo><mn>9</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2121" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2123" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2125" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2127" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2129" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2131" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2132" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2133" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p144"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2134" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>+</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa71">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2135" display="block"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa72">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2137" display="block"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p150"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2139" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2141" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2143" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>2</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mrow><mtext>−</mtext><mn>2</mn></mrow></msqrt><mo>−</mo><msqrt><mn>6</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2145" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt><mo>+</mo><msqrt><mn>8</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2147" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2149" display="inline"><mrow><msqrt><mrow><mtext>−</mtext><mn>15</mn></mrow></msqrt><mrow><mo>(</mo><mrow><msqrt><mn>3</mn></msqrt><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>10</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p162"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2151" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>2</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>2</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p164"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2152" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>5</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>5</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p166"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2153" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p168"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2155" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mn>2</mn><msqrt><mrow><mtext>−</mtext><mn>16</mn></mrow></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2157" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mo>+</mo><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2159" display="inline"><mrow><mrow><mo>(</mo><mrow><mtext>−</mtext><mn>1</mn><mo>+</mo><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mn>2</mn><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2161" display="block"><mrow><mfrac><mrow><mtext>−</mtext><mn>3</mn></mrow><mi>i</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2163" display="block"><mrow><mfrac><mn>5</mn><mi>i</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa87">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2165" display="block"><mrow><mfrac><mn>1</mn><mrow><mn>5</mn><mo>+</mo><mn>4</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa88">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2167" display="block"><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa89">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2169" display="block"><mrow><mfrac><mrow><mn>15</mn></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa90">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2171" display="block"><mrow><mfrac><mrow><mn>29</mn></mrow><mrow><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2173" display="block"><mrow><mfrac><mrow><mn>20</mn><mi>i</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2175" display="block"><mrow><mfrac><mrow><mn>10</mn><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa93">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2177" display="block"><mrow><mfrac><mrow><mn>10</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow><mrow><mn>3</mn><mo>−</mo><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa94">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2179" display="block"><mrow><mfrac><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa95">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2181" display="block"><mrow><mfrac><mrow><mn>5</mn><mo>+</mo><mn>10</mn><mi>i</mi></mrow><mrow><mn>3</mn><mo>+</mo><mn>4</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa96">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2183" display="block"><mrow><mfrac><mrow><mn>2</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mrow><mn>5</mn><mo>+</mo><mn>3</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa97">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2185" display="block"><mrow><mfrac><mrow><mn>26</mn><mo>+</mo><mn>13</mn><mi>i</mi></mrow><mrow><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa98">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2187" display="block"><mrow><mfrac><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa99">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2189" display="block"><mrow><mfrac><mrow><mn>3</mn><mo>−</mo><mi>i</mi></mrow><mrow><mn>2</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa100">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2191" display="block"><mrow><mfrac><mrow><mtext>−</mtext><mn>5</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow><mrow><mn>4</mn><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2193" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2195" display="block"><mrow><mfrac><mi>i</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa103">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2197" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mrow><mrow><mpadded height="1.5em"><mn>1</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>1</mn></mrow></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa104">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2199" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mrow><mrow><mpadded height="1.5em"><mn>1</mn><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>9</mn></mrow></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa105">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2201" display="block"><mrow><mfrac><mrow><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>6</mn></mrow></msqrt></mrow><mrow><mpadded height="1.7em"><msqrt><mrow><mn>18</mn></mrow></msqrt><mo>+</mo><msqrt><mrow><mtext>−</mtext><mn>4</mn></mrow></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa106">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2203" display="block"><mrow><mfrac><mrow><msqrt><mrow><mtext>−</mtext><mn>12</mn></mrow></msqrt></mrow><mrow><mpadded height="1.4em"><msqrt><mn>2</mn></msqrt><mo>−</mo><msqrt><mrow><mtext>−</mtext><mn>27</mn></mrow></msqrt></mpadded></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd03_qd02" start="107">
<p class="para" id="fwk-redden-ch05_s07_qs01_p218"><strong class="emphasis bold">Given that</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2205" display="inline"><mrow><msup><mi>i</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>i</mi><mi>n</mi></msup></mrow></mfrac></mrow></math></span> <strong class="emphasis bold">compute the following powers of</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2206" display="inline"><mi>i</mi><mo>.</mo></math></span></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p219"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2207" display="inline"><mrow><msup><mi>i</mi><mrow><mtext>−</mtext><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p221"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2209" display="inline"><mrow><msup><mi>i</mi><mrow><mtext>−</mtext><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p223"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2211" display="inline"><mrow><msup><mi>i</mi><mrow><mtext>−</mtext><mn>3</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p225"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2213" display="inline"><mrow><msup><mi>i</mi><mrow><mtext>−</mtext><mn>4</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd03_qd03" start="111">
<p class="para" id="fwk-redden-ch05_s07_qs01_p227"><strong class="emphasis bold">Perform the operations and simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p228"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2215" display="inline"><mrow><mn>2</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>i</mi><mrow><mo>(</mo><mrow><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p230"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2217" display="inline"><mrow><mi>i</mi><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mi>i</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>6</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p232"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2219" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p234"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2221" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>i</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p236"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2223" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p238"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2224" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa117">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p240"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2225" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa118">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p242"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2226" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>i</mi><msqrt><mn>5</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>i</mi><msqrt><mn>3</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa119">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p244"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2227" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mn>2</mn></msqrt><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><msqrt><mn>2</mn></msqrt><mo>+</mo><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa120">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p246"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2229" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>i</mi><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>i</mi><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa121">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2231" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa122">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch05_m2233" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa123">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p252"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2235" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa124">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p254"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2237" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa125">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p256">Show that both <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2239" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mi>i</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2240" display="inline"><mrow><mn>2</mn><mi>i</mi></mrow></math></span> satisfy <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2241" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa126">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p257">Show that both <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2242" display="inline"><mrow><mo>−</mo><mi>i</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2243" display="inline"><mi>i</mi></math></span> satisfy <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2244" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa127">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p258">Show that both <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2245" display="inline"><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2246" display="inline"><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></math></span> satisfy <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2247" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>13</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa128">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p259">Show that both <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2248" display="inline"><mrow><mn>5</mn><mo>−</mo><mi>i</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2249" display="inline"><mrow><mn>5</mn><mo>+</mo><mi>i</mi></mrow></math></span> satisfy <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2250" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>26</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa129">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p260">Show that 3, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2251" display="inline"><mrow><mo>−</mo><mn>2</mn><mi>i</mi></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2252" display="inline"><mrow><mn>2</mn><mi>i</mi></mrow></math></span> are all solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2253" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa130">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p261">Show that −2, <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2254" display="inline"><mrow><mn>1</mn><mo>−</mo><mi>i</mi></mrow></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2255" display="inline"><mrow><mn>1</mn><mo>+</mo><mi>i</mi></mrow></math></span> are all solutions to <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2256" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd04">
<h3 class="title">Part D: Discussion Board.</h3>
<ol class="qandadiv" id="fwk-redden-ch05_s07_qs01_qd04_qd01" start="131">
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa131">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p262">Research and discuss the history of the imaginary unit and complex numbers.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa132">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p263">How would you define <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2257" display="inline"><mrow><msup><mi>i</mi><mn>0</mn></msup></mrow></math></span> and why?</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa133">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p264">Research what it means to calculate the absolute value of a complex number <span class="inlineequation"><math xml:id="fwk-redden-ch05_m2258" display="inline"><mrow><mrow><mo>|</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi></mrow><mo>|</mo></mrow></mrow><mo>.</mo></math></span> Illustrate your finding with an example.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa134">
<div class="question">
<p class="para" id="fwk-redden-ch05_s07_qs01_p265">Explore the powers of <em class="emphasis">i</em>. Look for a pattern and share your findings.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch05_s07_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch05_s07_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2002" display="inline"><mrow><mn>9</mn><mi>i</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch05_s07_qs01_p07_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2006" display="inline"><mrow><mtext>−</mtext><mn>2</mn><mi>i</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch05_s07_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2010" display="inline"><mrow><mn>2</mn><mi>i</mi><msqrt><mn>5</mn></msqrt></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch05_s07_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>