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<title>Cubes and Cube Roots</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Cubes and Cube Roots</h1>
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<p align="center"><i>To understand cube roots, first we must understand
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cubes ...</i></p>
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<h2>How to Cube A Number</h2>
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<div align="center">
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<p>
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<span class="large">To <b>cube</b> a number, just use it in a multiplication <b>3 times</b> ... </span></p>
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</div>
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<h3>Example: What is 3 Cubed?</h3>
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<table border="0" align="center" cellpadding="4">
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<tr class="large">
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<td class="large">3 Cubed</td>
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<td class="large">=</td>
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<td height="180" valign="top"><img src="images/cube-3.svg" alt="cube 3x3x3" /></td>
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<td class="large"> </td>
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</tr>
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<tr class="large">
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<td class="large"> </td>
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<td class="large">=</td>
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<td align="center">3 × 3 × 3</td>
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<td class="large">= 27</td>
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</tr>
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</table>
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<p>Note: we write "3
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Cubed" as <span class="large">3<sup>3</sup></span><span class="large"><br />
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</span>(the little <sup>3</sup> means
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the number appears three times in multiplying)</p>
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<h2 align="left">Cubes From 0<sup>3</sup> to 6<sup>3</sup></h2>
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<table border="0" align="center" cellpadding="4" cellspacing="4">
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<tr>
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<td>0 cubed</td>
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<td>=</td>
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<td>0<sup>3</sup></td>
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<td>=</td>
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<td>0 × 0 × 0</td>
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<td>=</td>
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<td align="right">0</td>
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</tr>
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<tr>
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<td>1 cubed</td>
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<td>=</td>
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<td>1<sup>3</sup></td>
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<td>=</td>
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<td>1 × 1 × 1</td>
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<td>=</td>
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<td align="right">1</td>
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</tr>
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<tr>
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<td>2 cubed</td>
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<td>=</td>
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<td>2<sup>3</sup></td>
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<td>=</td>
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<td>2 × 2 × 2</td>
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<td>=</td>
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<td align="right">8</td>
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</tr>
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<tr>
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<td>3 cubed</td>
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<td>=</td>
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<td>3<sup>3</sup></td>
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<td>=</td>
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<td>3 × 3 × 3</td>
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<td>=</td>
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<td align="right">27</td>
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</tr>
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<tr>
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<td>4 cubed</td>
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<td>=</td>
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<td>4<sup>3</sup></td>
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<td>=</td>
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<td>4 × 4 × 4</td>
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<td>=</td>
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<td align="right">64</td>
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</tr>
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<tr>
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<td>5 cubed</td>
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<td>=</td>
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<td>5<sup>3</sup></td>
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<td>=</td>
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<td>5 × 5 × 5</td>
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<td>=</td>
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<td align="right">125</td>
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</tr>
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<tr>
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<td>6 cubed</td>
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<td>=</td>
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<td>6<sup>3</sup></td>
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<td>=</td>
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<td>6 × 6 × 6</td>
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<td>=</td>
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<td align="right">216</td>
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</tr>
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</table>
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<h2>Cube Root</h2>
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<p>A <b>cube root</b> goes the other direction:</p>
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<p class="large" align="center">3 cubed is 27, so the <b>cube root
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of 27 is 3</b></p>
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<table border="0" class="large" align="center" cellpadding="4" cellspacing="4">
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<tr valign="middle">
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<td class="huge">3</td>
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<td><img src="images/cube-cube-root.svg" alt="cube root direction" /></td>
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<td class="huge">27</td>
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</tr>
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</table>
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<p class="larger">The cube root of a number is ...<br />
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<span class="indent50px">... a special value that when <b>cubed</b> gives the original number.</span></p>
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<p class="larger">The cube root of <b>27</b> is ...<br />
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<span class="indent50px">... <b>3</b>, because <b>when 3 is cubed<b></b></b> you get <b>27</b>.</span></p>
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<table border="0" align="center">
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<tr>
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<td><img src="../algebra/images/tree-root.jpg" alt="tree root" width="67" height="118" /></td>
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<td> </td>
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<td><p>Note: 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>In this case the tree is "27", and the cube root is "3".</p></td>
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</tr>
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</table>
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<p>Here are some more cubes and cube roots: </p>
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<table border="0" align="center" cellpadding="4" cellspacing="4">
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<tr align="center">
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<td class="large" bgcolor="#FFFFFF" colspan="3"><img src="images/cube-cube-root.svg" alt="cube root direction" /></td>
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</tr>
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<tr align="center">
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<td class="large" bgcolor="#FFCCFF" width="33%">
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<div align="center">4</div> </td>
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<td class="large" bgcolor="#FFCCFF" width="33%"> </td>
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<td class="large" bgcolor="#FFCCFF" width="33%">
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<div align="center">64</div> </td>
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</tr>
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<tr align="center">
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<td class="large" bgcolor="#66FF99" width="33%">
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<div align="center">5</div> </td>
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<td class="large" bgcolor="#66FF99" width="33%"> </td>
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<td class="large" bgcolor="#66FF99" width="33%">
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<div align="center">125</div> </td>
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</tr>
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<tr align="center">
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<td class="large" bgcolor="#66CCCC" width="33%">
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<div align="center">6</div> </td>
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<td class="large" bgcolor="#66CCCC" width="33%"> </td>
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<td class="large" bgcolor="#66CCCC" width="33%">
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<div align="center">216</div> </td>
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</tr>
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</table>
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<br />
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<div class="example">
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<h3>Example: What is the Cube root of 125?</h3>
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<p>Well, we just happen to know that <b>125 = 5 × 5 × 5</b> (if you use
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5 three times in a multiplication you will get 125) ... </p>
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<p align="center"><b>... so the cube root of 125 is 5</b></p>
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</div>
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<h2> The Cube Root Symbol<br />
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</h2>
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<table border="0" align="center">
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<tr>
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<td align="center"><img src="images/cube-root-symbol.gif" width="30" height="57" alt="cube root symbol" /></td>
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<td width="20"> </td>
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<td width="320"><p>This is the special symbol that means "cube root",
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it is the <i><b>"radical"</b></i> symbol (used for square roots)
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with a little three to mean <b>cube</b> root.</p> </td>
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</tr>
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</table>
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<p>You can use it like this: <img src="images/cube-root-27.gif" alt="cube root 27 = 3" width="103" height="42" align="absmiddle" /> (we say "the cube root of 27 equals 3")</p>
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<p align="center"> </p>
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<h2>You Can Also Cube Negative Numbers</h2>
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<p>Have a look at this: </p>
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<div class="tbl">
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<div class="row"><span class="left">When we cube +5 we get +125:</span><span class="right">+5 × +5 × +5 = +125</span></div>
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<div class="row"><span class="left">When we cube −5 we get −125:</span><span class="right">−5 × −5 × −5 = −125</span></div>
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</div>
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<p align="center" class="large">So the <b>cube root</b> of −125 is<b></b> −5</p>
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<h2>Perfect Cubes</h2>
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<p>The Perfect Cubes are the cubes of the <a href="../whole-numbers.html">whole numbers</a>:</p>
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<div class="simple">
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<table align="center">
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<tr>
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<td> </td>
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<td align="right"><b>Perfect<br>
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Cubes</b></td>
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</tr>
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<tr><td>0</td><td align="right"><b>0</b></td></tr>
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<tr><td>1</td><td align="right"><b>1</b></td></tr>
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<tr><td>2</td><td align="right"><b>8</b></td></tr>
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<tr><td>3</td><td align="right"><b>27</b></td></tr>
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<tr><td>4</td><td align="right"><b>64</b></td></tr>
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<tr><td>5</td><td align="right"><b>125</b></td></tr>
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<tr><td>6</td><td align="right"><b>216</b></td></tr>
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<tr><td>7</td><td align="right"><b>343</b></td></tr>
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<tr><td>8</td><td align="right"><b>512</b></td></tr>
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<tr><td>9</td><td align="right"><b>729</b></td></tr>
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<tr><td>10</td><td align="right"><b>1000</b></td></tr>
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<tr><td>11</td><td align="right"><b>1331</b></td></tr>
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<tr><td>12</td><td align="right"><b>1728</b></td></tr>
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<tr><td>13</td><td align="right"><b>2197</b></td></tr>
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<tr><td>14</td><td align="right"><b>2744</b></td></tr>
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<tr><td>15</td><td align="right"><b>3375</b></td></tr>
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</table>
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</div>
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<p>It is easy to work out the cube root of a perfect cube, but it
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is <b>really hard</b> to work out other cube roots.</p>
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<div class="example">
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<h3>Example: what is the cube root of 30?</h3>
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<p>Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4. </p>
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<ul>
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<li>Let's try 3.5: <i>3.5 × 3.5 × 3.5 = 42.875</i></li>
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<li>Let's try 3.2: <i>3.2 × 3.2 × 3.2 = 32.768</i></li>
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<li>Let's try 3.1: <i>3.1 × 3.1 × 3.1 = 29.791</i></li>
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</ul>
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<p>We are getting closer, but very slowly ... at this point, I get out my calculator and it says: </p>
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<div align="center">
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<p><i>3.1072325059538588668776624275224</i>...</p>
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</div>
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<p>... but the digits just go on and on, without any pattern. So even
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the calculator's answer is <b>only an <i>approximation</i> ! </b><br />
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<i><br />
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(Further reading: these kind of numbers are called <a href="../surds.html">surds</a> which are a special type of <a href="../irrational-numbers.html">irrational number</a>)</i> <br />
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</p>
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</div><p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(316, 2007, 2008, 5081, 5084, 3157, 3158, 5082, 5083, 5085);</script>
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</div>
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<div class="related">
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<a href="../square-root.html">Squares and Square Roots</a>
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<a href="../surds.html">Surds</a>
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<a href="../scientific-calculator.html">Scientific Calculator</a>
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<a href="../algebra/index.html">Algebra Index</a> </div>
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Copyright © 2017 MathsIsFun.com
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