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<title>Rationalize the Denominator</title>
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<h1 align="center">Rationalize the Denominator</h1>
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<table border="0">
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<tr>
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<td><img src="images/rationalize-denom1.svg" alt="1/(square root of 2) becomes (square root of 2)/2" /></td>
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<td width="30"> </td>
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<td><b>"Rationalizing the denominator"</b> is when we move a root (like a <a href="../square-root.html">square root</a> or <a href="../numbers/cube-root.html">cube root</a>) from the bottom of a fraction to the top. </td>
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<td> </td>
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</tr>
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</table>
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<h2>Oh No! An Irrational Denominator!</h2>
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<p class="words">The bottom of a fraction is called the <b>denominator</b>. <br />
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Numbers like 2 and 3 are <a href="../rational-numbers.html">rational</a>.<br>
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But many roots, such as √2 and √3, are <a href="../irrational-numbers.html">irrational</a>.</p>
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<div class="example">
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<h3>Example: <img src="images/1-root2.gif" alt="1 divide root 2" width="29" height="46" align="absmiddle" /> has an Irrational Denominator</h3>
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<p class="center"><img src="images/rationalize-denom2.svg" alt="square root of 2 is irrational and is denominator" /></p>
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</div>
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<table border="0" align="center">
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<tr>
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<td>
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</td>
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</tr>
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</table>
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<p align="center" class="large">To be in "simplest form" the denominator should <b>not</b> be irrational! </p>
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<p align="center" class="larger">Fixing it (by making the denominator rational)<br>
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is called "<b>Rationalizing the Denominator</b>"</p>
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<p>Note: there is nothing <b>wrong</b> with an irrational denominator, it still works. But it is not "simplest form" and so <b>can cost you marks</b>. </p>
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<p>And removing them may help you solve an equation, so you should learn how.</p>
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<p>So ... how do we do it?</p>
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<h2>1. Multiply Both Top and Bottom by a Root</h2>
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<p>Sometimes we can just multiply both top and bottom by a root:</p>
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<div class="example">
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<h3>Example: <img src="images/1-root2.gif" alt="1 divide root 2" width="29" height="46" align="absmiddle" /> has an Irrational Denominator. Let's fix it.</h3>
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<p>Multiply top and bottom by the square root of 2, because: √2 × √2 = 2: </p>
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<p align="center"><img src="images/1-root2-rationalize.png" alt="rationalized" width="145" height="52" align="absmiddle" /></p>
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<p>Now the denominator has a <a href="../rational-numbers.html">rational number</a> (=2). Done!</p>
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</div>
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<p>Note: It is ok to have an irrational number in the top (numerator) of a fraction.</p>
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<h2>2. Multiply Both Top and Bottom by the Conjugate</h2>
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<p>There is another special way to move a square root from the bottom of a fraction to the top ... we <b>multiply both top and bottom </b>by the<b> <a href="conjugate.html">conjugate</a> of the denominator</b>.</p>
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<div class="def">
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<p>The conjugate is where we <b>change the sign in the middle</b> of two terms:</p>
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<table border="0" align="center">
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<tr align="center">
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<th class="larger">Example Expression</th>
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<th width="20" class="larger"> </th>
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<th class="larger">Its Conjugate</th>
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</tr>
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<tr align="center">
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<td class="larger">x<sup>2</sup> − 3</td>
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<td class="larger"> </td>
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<td class="larger">x<sup>2</sup> <span class="hi">+</span> 3</td>
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</tr>
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</table>
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<br />
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<table border="0" align="center">
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<tr align="center">
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<th class="larger">Another Example</th>
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<th width="20" class="larger"> </th>
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<th class="larger">Its Conjugate</th>
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</tr>
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<tr align="center">
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<td class="larger">a + b<sup>3</sup></td>
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<td class="larger"> </td>
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<td class="larger">a <span class="hi">−</span> b<sup>3</sup></td>
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</tr>
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</table>
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</div>
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<p>It works because when we multiply something by its conjugate we get <b>squares</b> like this:</p>
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<p class="center large">(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup> </p>
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<p>Here is how to do it:</p>
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<div class="example">
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<h3><b>Example: </b><b> </b>here is a fraction with an "irrational denominator":<br>
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<br>
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<div class="center large"><span class="intbl">
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<em>1</em>
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<strong>3−√2</strong>
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</span></div>
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</h3>
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<p>How can we move the square root of 2 to the top?</p>
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<p>We can <i><b>multiply both top and bottom by 3+√2 (the conjugate of 3−√2)</b></i>, which won't change the value of the fraction:</p>
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<p class="center large"><span class="intbl">
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<em>1</em>
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<strong>3−√2</strong>
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</span> × <span class="intbl">
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<em>3+√2</em>
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<strong>3+√2</strong>
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</span> = <span class="intbl">
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<em>3+√2</em>
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<strong>3<sup>2</sup>−(√2)<sup>2</sup></strong>
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</span> = <span class="intbl">
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<em>3+√2</em>
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<strong>7</strong>
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</span></p>
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<p align="center">(Did you see that we used <b>(a+b)(a−b) = a<sup>2</sup> − b<sup>2</sup></b> in the denominator?)</p>
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<p> </p>
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<p><i>Use your calculator to work out the value before and after ... is it the same?</i></p>
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</div>
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<p>There is another example on the page <a href="../calculus/limits-evaluating.html">Evaluating Limits</a> (advanced topic) where I move a square root from the top to the bottom.</p>
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<h3>Useful</h3>
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<p>So try to remember these little tricks, it may help you solve an equation one day! </p>
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<p> </p>
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<script type="text/javascript">getQ(355, 2248, 2249, 2250, 2251, 3196, 3197, 3198, 3199, 97);</script>
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</div>
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<div class="related">
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<a href="../rational-numbers.html">Rational Number</a><a href="index.html">Algebra Index</a> </div>
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