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435 lines
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<title>nth Roots</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">nth Root</h1>
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<p class="center"><i>The "nth Root" used <b>n times</b> in a <b>multiplication</b> gives the original value</i></p>
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<h2>" nth ? "</h2>
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<p class="center larger"><b>1</b>st, <b>2</b>nd, <b>3</b>rd, <b>4</b>th, <b>5</b>th, ... <b>n</b>th ...</p>
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<p>Instead of talking about the "4th", "16th", etc, we can just say the "<span class="large"><i><b>n</b>th</i> </span>".</p>
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<h2>The nth Root</h2>
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<ul>
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<li>The "2nd" root is the <a href="../square-root.html">square root</a></li>
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<li>The "3rd" root is the <a href="cube-root.html">cube root</a></li>
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<li>etc!</li>
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</ul><br>
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<div class="simple">
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<table style="border: 0;">
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<tbody>
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<tr>
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<td class="largest" align="center">2</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center; width:260px;">
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<span class="nthroot"> </span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:170%;">×</span>
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<span class="nthroot"> </span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:140%;"> = a</span>
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</td>
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<td> </td>
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<td>The <b>square root</b> used <b>two</b> times in a multiplication gives the original value.</td>
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</tr>
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<tr>
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<td class="largest" align="center">3</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:170%;">×</span>
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:170%;">×</span>
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:140%;"> = a</span>
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</td>
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<td> </td>
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<td> The <b>cube root</b> used <b>three</b> times in a multiplication gives the original value.</td>
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</tr>
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<tr>
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<td style="text-align:center;"><span class="vertellip"></span><br>
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</td>
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<td> </td>
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<td style="text-align:center;"><span class="vertellip"></span><br>
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</td>
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<td> </td>
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<td style="text-align:center;"><span class="vertellip"></span><br>
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</td>
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</tr>
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<tr>
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<td class="largest" align="center">n</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:170%;">×</span>
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:170%;">×</span>
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<span style="font-size:130%;">...</span>
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<span style="font-size:170%;">×</span>
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:130%;"> = a</span><br>
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<i>(n of them)</i>
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</td>
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<td> </td>
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<td>The <b>nth root</b> used <b>n</b> times in a multiplication gives the original value.</td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="def">
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<p class="center larger">So it is the <b>general</b> way of talking about roots<br>
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(so it could be 2nd, or 9th, or 324th, or whatever)</p>
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</div>
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<h2>The nth Root Symbol</h2>
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<p style="float:left; margin: 0 30px 5px 0;"> <img src="images/nth-root-symbol.gif" alt="nth root symbol" height="54" width="30"></p>
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<p>This is the special symbol that means "nth root",
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it is the <i><b>"radical"</b></i> symbol (used for square roots) with a little <b>n</b> to mean <b>nth</b> root.</p>
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<h2>Using it</h2>
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<p>We could use the nth root in a question like this:</p>
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<div class="example">
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<p>Question: What is "n" in this equation?</p>
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<p class="center larger"><span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">625</span> = 5</p>
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<p>Answer: I just happen to know that <b>625 = 5<sup>4</sup></b> , so the <b>4</b>th root of 625 must be 5:</p>
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<p class="center larger"><span class="nthroot">4</span><span style="font-size:120%;">√</span><span class="overline">625</span> = 5</p>
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</div>
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<p>Or we could use "n" because we want to say general things:</p>
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<div class="example">
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<p>Example: When <b>n</b> is odd then
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<span class="larger">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a<sup>n</sup></span> = a
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</span>
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(we talk about this later).</p>
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</div>
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<h2>Why "Root" ... ?</h2>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td><img src="../algebra/images/tree-root.jpg" alt="tree root" height="118" width="67"></td>
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<td>
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<p>When you see "root" think</p>
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<p><i>"I know the tree</i><i>, but what is the root that produced it?</i> "</p>
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<p>Example: in <b>√9 = 3</b> the "tree" is <b>9</b> , and the root is <b>3</b> .</p></td>
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</tr>
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</tbody></table>
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<h2>Properties</h2>
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<p>Now we know what an nth root is, let us look at some properties:</p>
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<h3>Multiplication and Division</h3>
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<p>We can "pull apart" multiplications under the root sign like this:</p>
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<p class="center"><span class=" larger">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">ab</span> =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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×
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">b</span>
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</span><br>
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(<i>Note: if n is even then a and b must both be ≥ 0)</i></p>
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<p>This can help us simplify equations in algebra, and also make some calculations easier:</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center"><span class=" larger">
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">128</span> =
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">64×2</span> =
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">64</span>
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×
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">2</span> =
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4<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">2</span>
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</span></p>
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<p>
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So the cube root of 128 simplifies to 4 times the cube root of 2.
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</p>
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</div>
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<p>It also works for division:</p>
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<p class="center"><span class=" larger">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a/b</span> =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span>
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<span style="font-size:140%;">/</span>
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">b</span>
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</span><br>
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(<i>a≥0 and b>0)<br>
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Note that b cannot be zero, as we can't divide by zero</i></p>
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<div class="example">
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<p>Example:</p>
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<p class="center"><span class=" larger">
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">1/64</span> =
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">1</span>
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<span style="font-size:140%;">/</span>
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<span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">64</span> =
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1/4
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</span></p>
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<p>
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So the cube root of 1/64 simplifies to just one quarter.
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</p>
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</div>
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<h3>Addition and Subtraction</h3>
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<p>But we <b>cannot</b> do that kind of thing for additions or subtractions!</p>
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<p class="center larger"><img src="../images/style/no.svg" alt="no!" style="vertical-align:middle;" height="30" width="30">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a + b</span>
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<span style="font-size:140%;">≠</span>
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span> +
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">b</span></p>
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<p class="center larger"><img src="../images/style/no.svg" alt="no!" style="vertical-align:middle;" height="30" width="30">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a − b</span>
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<span style="font-size:140%;">≠</span>
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span> −
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">b</span></p>
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<p class="center larger"><img src="../images/style/no.svg" alt="no!" style="vertical-align:middle;" height="30" width="30">
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a<sup>n</sup> + b<sup>n</sup></span>
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<span style="font-size:140%;">≠</span> a + b</p>
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<div class="example">
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<p>Example: <a href="../pythagoras.html">Pythagoras' Theorem</a> says</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="../geometry/images/triangle-abc.svg" alt="Right angled triangle" height="109" width="189"></td>
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<td> </td>
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<td><span class="large">a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup></span></td>
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</tr>
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</tbody></table>
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<p>So we calculate c like this:</p>
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<p class="center larger">c = <span style="font-size:120%;">√</span><span class="overline">a<sup>2</sup> + b<sup>2</sup></span></p>
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<p>Which is <b>not</b> the same as <b>c = a + b</b> , right?</p>
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</div>
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<p>It is an easy trap to fall into, so beware.</p>
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<p>It also means that, unfortunately, additions and subtractions can be hard to deal with when under a root sign.</p>
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<p> </p>
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<h3>Exponents vs Roots</h3>
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<p>An exponent on one side of "=" can be turned into a root on the other side of "=":</p>
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<p class="center larger">If <b>a<sup>n</sup> = b</b> then <b>a =
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<span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">b</span></b></p>
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<p>Note: when n is even then b must be ≥ 0</p>
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<div class="example">
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<h3>Example:</h3>
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<p class="center larger"><b>5<sup>4</sup> = 625</b> so <b>5 = <span class="nthroot">4</span><span style="font-size:120%;">√</span><span class="overline">625</span></b></p>
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</div>
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<p> </p>
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<h3>nth Root of a-to-the-nth-Power</h3>
|
||
<p>When a value has an <b><a href="../exponent.html">exponent</a> of n</b> and we take the <b>nth root</b> we <b>get the value back again</b> ...</p>
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p class="large">... when a is <b>positive</b> (or zero):</p></td>
|
||
<td><br></td>
|
||
<td style="width:10px;"> </td>
|
||
<td><img src="images/nth-root-n-a-n.svg" alt="nth root a^n" height="34" width="90"></td>
|
||
<td style="width:30px;"> </td>
|
||
<td><i>(when <b>a ≥ 0</b> )</i> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="example">
|
||
<p>Example: <img src="images/nth-root-3-2-3.svg" alt="root examples" style="vertical-align:middle; margin-left:20px;" height="34" width="91"></p>
|
||
</div>
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td>
|
||
<p class="large">... or when the <b>exponent is odd</b> :</p></td>
|
||
<td><br></td>
|
||
<td style="width:10px;"> </td>
|
||
<td><img src="images/nth-root-n-a-n.svg" alt="nth root a^n" height="34" width="90"></td>
|
||
<td style="width:30px;"> </td>
|
||
<td><i> (when <b>n is odd</b> )</i> </td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="example">
|
||
<p>Example:<img src="images/nth-root-3-m2-3.svg" alt="root examples" style="vertical-align:middle; margin-left:20px;" height="34" width="114"></p>
|
||
</div>
|
||
<p class="larger">... but when <b>a is negative</b> and the <b>exponent is even</b> we get this:</p>
|
||
<p class="center"><img src="images/nth-root-2-m3-2.svg" alt="Square root of square" height="34" width="117"></p>
|
||
<p class="center">Did you see that −3 became +3 ?</p>
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="large">... so we must do this:</td>
|
||
<td><br></td>
|
||
<td style="width:10px;"> </td>
|
||
<td><img src="images/nth-root-n-a-n-abs.svg" alt="nth root a^n = abs(a)" height="34" width="110"></td>
|
||
<td style="width:30px;"> </td>
|
||
<td><i>(when <b>a < 0</b> and <b>n is even</b> )</i><br>
|
||
</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The <b class="large">|a|</b> means the <a href="absolute-value.html">absolute value</a> of <b>a</b>, in other words any negative becomes a positive.</p>
|
||
<div class="example">
|
||
<p>Example:<img src="images/nth-root-4-m2-4.svg" alt="4th root example" style="vertical-align:middle; margin-left:20px;" height="34" width="184"></p>
|
||
</div>
|
||
<p>So that is something to be careful of! Read more at <a href="../algebra/exponents-squaring-negative.html">Exponents of Negative Numbers</a></p>
|
||
<p>Here it is in a little table:</p>
|
||
<div class="simple">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;">
|
||
<th> </th>
|
||
<th width="150">n is odd</th>
|
||
<th width="150">n is even</th>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<th>a ≥ 0</th>
|
||
<td style="width:150px;"><img src="images/nth-root-n-a-n.svg" alt="nth root a^n" height="34" width="90"></td>
|
||
<td style="width:150px;"><img src="images/nth-root-n-a-n.svg" alt="nth root a^n" height="34" width="90"></td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<th>a < 0</th>
|
||
<td style="width:150px;"><img src="images/nth-root-n-a-n.svg" alt="nth root a^n" height="34" width="90"></td>
|
||
<td width="150" bgcolor="#FFFFCC"><img src="images/nth-root-n-a-n-abs.svg" alt="nth root a^n = abs(a)" height="34" width="110"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h3>nth Root of a-to-the-mth-Power</h3>
|
||
<p>What happens when the exponent and root are different values (<b>m</b> and <b>n</b>)?</p>
|
||
<p>Well, we are allowed to change the order like this:</p>
|
||
<p class="center larger"><span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a<sup>m</sup></span> =
|
||
<span style="font-size:130%;">(</span><span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span> <span style="font-size:130%;">)<sup>m</sup></span></p>
|
||
<p>So this: nth root of (a to the power m)<br>
|
||
becomes (nth root of a) to the power m</p>
|
||
<div class="example">
|
||
|
||
<h3>Example:</h3>
|
||
<p class="center larger"><span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">27<sup>2</sup></span> =
|
||
<span style="font-size:130%;">(</span><span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">27</span> <span style="font-size:130%;">)<sup>2</sup></span><br>
|
||
= 3<sup>2</sup><br>
|
||
= 9</p>
|
||
<p>Easier than squaring 27 then taking a cube root, right?</p>
|
||
</div>
|
||
<p><br></p>
|
||
<p>But there is an even <b>more powerful method</b> ... we can combine the exponent and root to make a new exponent, like this:</p>
|
||
<p class="center larger"><span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a<sup>m</sup></span> =
|
||
<span style=" transform: scale(1.3); display:inline-block;">a</span><sup><span class="intbl"><em>m</em><strong>n</strong></span></sup></p>
|
||
<p>The new exponent is the fraction <span class="intbl"><em>m</em><strong>n</strong></span> which may be easier to solve.</p>
|
||
<div class="example">
|
||
|
||
<h3>Example:</h3>
|
||
<p class="center larger"><span class="nthroot">3</span><span style="font-size:120%;">√</span><span class="overline">4<sup>6</sup></span>
|
||
= <span style=" transform: scale(1.4); display:inline-block;">4</span><sup><span class="intbl"><em>6</em><strong>3</strong></span></sup><br>
|
||
= 4<sup>2</sup><br>
|
||
= 16</p>
|
||
</div>This works because the <b>nth root</b> is the same as an <b>exponent of (1/n)</b>
|
||
<p class="center larger"><span class="nthroot">n</span><span style="font-size:120%;">√</span><span class="overline">a</span> =
|
||
<span style=" transform: scale(1.3); display:inline-block;">a</span><sup><span class="intbl"><em>1</em><strong>n</strong></span></sup></p>
|
||
<div class="example">
|
||
<h3>Example:</h3>
|
||
<p class="center larger"><span class="nthroot">2</span><span style="font-size:120%;">√</span><span class="overline">9</span>
|
||
= <span style=" transform: scale(1.4); display:inline-block;">9</span><sup><span class="intbl"><em>1</em><strong>2</strong></span></sup>
|
||
= 3</p>
|
||
</div>
|
||
<p>You might like to read about <a href="../algebra/exponent-fractional.html">Fractional Exponents</a> to find out why!</p>
|
||
<p> </p>
|
||
<div class="questions">318, 2055, 319, 317, 1087, 2056, 1088, 2057, 3159, 3160</div>
|
||
|
||
<div class="related">
|
||
<a href="../square-root.html">Squares and Square Roots</a>
|
||
<a href="../surds.html">Surds</a>
|
||
<a href="../scientific-calculator.html">Scientific Calculator</a>
|
||
<a href="../algebra/index.html">Algebra Index</a>
|
||
</div>
|
||
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