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151 lines
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HTML
151 lines
7.2 KiB
HTML
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<title>Simplifying Square Roots</title>
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<h1 class="center">Simplifying Square Roots</h1>
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<p>To simplify a square root: make the number inside the square root as small as possible (but still a <a href="../whole-numbers.html">whole number</a>):</p>
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<div class="example">
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<h3>Example: √12 is simpler as 2√3</h3>
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<p>Get your calculator and check if you want: they are both the same value!</p>
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</div>
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<p>Here is the rule: when <span class="center"> a and b are not negative</span></p>
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<p class="center"><img src="images/root-ab-root-a-root-b.gif" alt="√(ab) = √a × √b" width="256" height="42" /></p>
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<p>And here is how to use it:</p>
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<div class="example">
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<h3>Example: simplify √12</h3>
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<p>12 is 4 times 3:</p>
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<div class="so">√12 = √(4 × 3)</div>
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<p>Use the rule:</p>
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<div class="so">√(4 × 3) = √4 × √3</div>
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<p>And the square root of 4 is 2:</p>
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<div class="so">√4 × √3 = 2√3</div>
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<p>So √12 is simpler as 2√3</p>
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</div>
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<p>Another example:</p>
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<div class="example">
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<h3>Example: simplify √8</h3>
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<div class="so">√8 = √(4×2) = √4 × √2 = 2√2 </div>
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<p>(Because the square root of 4 is 2)</p>
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</div>
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<p>And another: </p>
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<div class="example">
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<h3>Example: simplify √18</h3>
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<p class="so">√18 = √(9 × 2) = √9 × √2 = 3√2</p>
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</div>
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<p>It often helps to <a href="factors-multiples.html">factor</a> the numbers (into <a href="../prime-factorization.html">prime numbers</a> is best): </p>
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<div class="example">
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<h3>Example: simplify √6 × √15</h3>
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<p>First we can combine the two numbers:</p>
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<div class="so">√6 × √15 = √(6 × 15) </div>
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<p>Then we factor them:</p>
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<div class="so">√(6 × 15) = √(2 × 3 × 3 × 5) </div>
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<p>Then we see two 3s, and decide to "pull them out":</p>
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<div class="so">√(2 × 3 × 3 × 5) = √(3 × 3) × √(2 × 5) = 3√10</div>
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</div>
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<h2>Fractions</h2>
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<p>There is a similar rule for fractions:</p>
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<p class="center"><img src="images/root-a-on-b-root-a-root-b.gif" width="186" height="82" alt="root a / root b = root (a / b)" /></p>
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<div class="example">
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<h3>Example: simplify √30 / √10</h3>
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<p>First we can combine the two numbers:</p>
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<div class="so">√30 / √10 = √(30 / 10)</div>
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<p>Then simplify:</p>
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<div class="so">√(30 / 10) = √3 </div>
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</div>
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<h2>Some Harder Examples</h2>
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<div class="example">
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<h3>Example: simplify <span class="intbl"><em>√20 × √5</em><strong>√2</strong></span></h3>
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<p>See if you can follow the steps:</p>
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<div class="so"><span class="intbl"><em>√20 × √5</em><strong>√2</strong></span></div>
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<div class="so"><span class="intbl"><em>√(2 × 2 × 5) × √5</em><strong>√2</strong></span></div>
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<div class="so"><span class="intbl"><em>√2 × √2 × √5 × √5</em><strong>√2</strong></span></div>
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<div class="so">√2 × √5 × √5</div>
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<div class="so">√2 × 5</div>
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<div class="so">5√2</div>
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</div>
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<div class="example">
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<h3>Example: simplify 2√12 + 9√3</h3>
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<p>First simplify 2√12:</p>
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<div class="so">2√12 = 2 × 2√3 = 4√3</div>
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<p>Now both terms have √3, we can add them:</p>
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<div class="so">4√3 + 9√3 = (4+9)√3 = 13√3</div>
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</div>
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<h2>Surds</h2>
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<p>Note: a root we <b>can't</b> simplify further is called a <a href="../surds.html">Surd</a>. So √3 is a surd. But √4 = 2 is not a surd.</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(8805, 8806, 8807, 8808, 8809, 8810, 8811, 8812, 8813, 8814);</script>
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</div>
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<div class="related">
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<a href="../algebra/index.html">Algebra Index</a>
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<a href="index.html">Numbers Index</a>
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