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<h3 id="title" class="align-center">Al Azhar 9th Grade Lesson Notes</h3>
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<h1 class="banner-text" id="banner-txt">Root</h1>
<h3 class="banner-text" id="banner-txt">Chapter 1</h3>
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<h1 id="definition">Definition</h1>
Root is the inverse of exponent. For example;<br>
<ul>
<li>4<sup>2</sup> = 16, <b>so <span class="sroot">16</span> = 4</b></li>
<li>5<sup>2</sup> = 25, <b>so <span class="sroot">25</span> = 5</b></li>
<li>10<sup>2</sup> = 100, <b>so <span class="sroot">100</span> = 10</b></li>
</ul>
<h1 id="simplification">Simplifying Square Roots</h1>
You can simplify a root by factoring the root number another number that is rational. Or in other words, divide the root number by the perfect squares. For example;
<ul>
<li>
<strong>Simplify <span class="sroot">60</span></strong><br>
<span class="sroot">60</span> = <span class="sroot">4 × 15</span> = 2<span class="sroot">15</span><br>
</li>
<li>
<strong>Simplify <span class="sroot">48</span></strong><br>
<span class="sroot">48</span> = <span class="sroot">16 × 3</span> = 4<span class="sroot">3</span><br>
</li>
<li>
<strong>Simplify <span class="sroot">125</span></strong><br>
<span class="sroot">125</span> = <span class="sroot">25 × 5</span> = 5<span class="sroot">5</span><br>
</li>
</ul>
<h1 id="root-operations">Root Operations</h1>
<h2 id="root-operations--addition">Addition</h2>
Roots with coefficient can be added with another coefficient that have the same roots. <br>
<strong>a<span class="sroot">c</span> + b<span class="sroot">c</span> = (a + b)<span class="sroot">c</span></strong>
<ul>
<li>3<span class="sroot">2</span> + 4<span class="sroot">2</span> = (3 + 4)<span class="sroot">2</span> = <strong>7<span class="sroot">2</span></strong></li>
<li><span class="sroot">5</span> + 3<span class="sroot">5</span> = (1 + 3)<span class="sroot">5</span> = <strong>4<span class="sroot">5</span></strong></li>
</ul>
<h2 id="root-operations--subtraction">Subtraction</h2>
Roots with coefficient can be subtracted with another coefficient that have the same roots. <br>
<strong>a<span class="sroot">c</span> - b<span class="sroot">c</span> = (a - b)<span class="sroot">c</span></strong>
<ul>
<li>7<span class="sroot">3</span> - 4<span class="sroot">3</span> = (7 - 4)<span class="sroot">3</span> = <strong>3<span class="sroot">2</span></strong></li>
<li>10<span class="sroot">6</span> - 5<span class="sroot">6</span> = (10 - 5)<span class="sroot">6</span> = <strong>5<span class="sroot">6</span></strong></li>
</ul>
<h2 id="root-operations--multiplication">Multiplication</h2>
Roots can be multiplied with any other roots. If the root is the same as the other multiplied root, the root can be deleted (if you want to be faster). <br>
<strong>a<span class="sroot">c</span> × b<span class="sroot">d</span> = (a × b)<span class="sroot">c × d</span></strong>
<ul>
<li><span class="sroot">2</span> × <span class="sroot">5</span> = <strong><span class="sroot">10</span></strong></li>
<li>3<span class="sroot">2</span> × 4<span class="sroot">3</span> = (3 × 4)<span class="sroot">2 × 3</span> = <strong>12<span class="sroot">6</span></strong></li>
<li>2<span class="sroot">5</span> × 3<span class="sroot">5</span> = (2 × 3)(5) = 6(5) = <strong>30</strong></li>
</ul>
<h2 id="root-operations--division">Division</h2>
Roots can be divided with any other roots. <br>
<strong>a<span class="sroot">c</span> ÷ b<span class="sroot">d</span> = (a ÷ b)<span class="sroot">c ÷ d</span></strong>
<ul>
<li>
8<span class="sroot">6</span> ÷ 4<span class="sroot">3</span> = (8 ÷ 4)<span class="sroot">6 ÷ 3</span> = <strong>2<span class="sroot">2</span></strong>
</li>
</ul>
<h2 id="rationalising">Rationalising Roots</h2>
<p>We can rationalise or simplify a fraction with a root denominator. To rationalise a root, multiply the fraction with a fraction of opposite roots. For example:</p>
<p><b>Example 1:</b></p>
<span id="id1"></span>
<script defer>
var element = document.getElementById("id1");
katex.render(String.raw`\dfrac{3}{\sqrt{5}}\\
= \dfrac{3}{\sqrt{5}} \times \dfrac{\sqrt{5}}{\sqrt{5}}\\
= \dfrac{3\sqrt{5}}{5}
`, element, {
throwOnError: false
});
</script>
<p><b>Example 2:</b></p>
<span id="id2"></span>
<script defer>
var element = document.getElementById("id2");
katex.render(String.raw`\dfrac{2}{3\sqrt{6}}\\
= \dfrac{2}{3\sqrt{6}} \times \dfrac{\sqrt{6}}{\sqrt{6}}\\
= \dfrac{2\sqrt{6}}{3 \cdot 6}\\
= \dfrac{2\sqrt{6}}{18}\\
= \dfrac{1}{9}\sqrt{6}
`, element, {
throwOnError: false
});
</script>
<p><b>Example 3:</b></p>
<span id="id3"></span>
<script defer>
var element = document.getElementById("id3");
katex.render(String.raw`\dfrac{8}{3 + \sqrt{7}}\\
= \dfrac{8}{3 + \sqrt{7}} \times \dfrac{3 - \sqrt{7}}{3 - \sqrt{7}}\\
= \dfrac{8(3 \cdot \sqrt{7})}{(3 \cdot \sqrt{7})(3 \cdot \sqrt{7})}\\
= \dfrac{8(3 \cdot \sqrt{7})}{9 - 3\sqrt{7} + 3\sqrt{7} - 7}\\
= \dfrac{8(3 \cdot \sqrt{7})}{9 - 7}\\
= \dfrac{8(3 \cdot \sqrt{7})}{2}
`, element, {
throwOnError: false
});
</script>
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