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<title>Variables with Exponents - How to Multiply and Divide them</title>
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<h1 align="center">Variables with Exponents</h1>
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<h3 align="center">How to Multiply and Divide them</h3>
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<h2>What is a Variable with an Exponent?</h2>
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<p style="float:left; margin: 0 20px 5px 0;"><img src="images/exponent-variable.svg" alt="variable x with exponent 2" /></p>
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<p>A <b>Variable</b> is a symbol for a number we don't know yet.
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It is usually a letter like x or y.</p>
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<p>An <a href="../exponent.html">exponent</a> (such as the 2 in <b>x<sup>2</sup></b>) says how many times
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to use the variable in a multiplication.</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: <b>y<sup>2</sup> = yy</b> </h3>
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<p>(<b>yy</b> means <b>y <i>multiplied by</i> y</b>, because in Algebra putting two letters next to each other means to multiply them)</p>
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<p>Likewise<b> z<sup>3</sup> = zzz</b> and <b>x<sup>5</sup> = xxxxx</b></p>
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</div>
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<h2>Exponents of 1 and 0</h2>
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<h3>Exponent of 1</h3>
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<p>When the exponent is 1, we just have the variable itself (example <b>x<sup>1</sup></b> = <b>x</b>)</p>
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<p>We usually don't write the "1", but it sometimes helps to remember that <b>x</b> is also <b>x<sup>1</sup></b></p>
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<h3>Exponent of 0</h3>
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<p>When the exponent is 0, we are not multiplying by anything and the answer is just "1" <br>
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(example <b>y<sup>0</sup></b> = <b>1</b>)</p>
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<h2>Multiplying Variables with Exponents</h2>
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<p>So, how do we multiply this:</p>
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<p align="center" class="large">(y<sup>2</sup>)(y<sup>3</sup>)</p>
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<p>We know that <b>y<sup>2</sup> = yy</b>, and <b>y<sup>3</sup> = yyy</b> so let us write out all the multiplies: </p>
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<p align="center" class="large">y<sup>2</sup> y<sup>3</sup> = yy yyy</p>
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<p> That is 5 "y"s multiplied together, so the new exponent must be 5:</p>
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<p align="center" class="large">y<sup>2</sup> y<sup>3</sup> = y<sup>5</sup></p>
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<p class="larger">But why <b>count</b> the "y"s when the exponents already tell us how many?</p>
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<p>The exponents tell us there are two "y"s multiplied by 3 "y"s for a total of 5 "y"s:</p>
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<p align="center"><span class="large">y<sup>2</sup> y<sup>3</sup></span> = <span class="large">y<sup>2+3</sup></span> = <span class="large">y<sup>5</sup></span></p>
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<p>So, the simplest method is to just <b>add the exponents</b>!</p>
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<p>(Note: this is one of the <a href="exponent-laws.html">Laws of Exponents</a>)</p>
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<h2>Mixed Variables</h2>
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<p>When we have a mix of variables, just add up the exponents for each, like this (press play):</p>
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<script type="text/javascript">playerMain("images/variables-exponents-b.txt",600,120);</script>
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<p>(Remember: a variable without an exponent really has an exponent of 1, example: <b>y</b> is <b>y<sup>1</sup></b>)</p>
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<h2>With Constants</h2>
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<p>There will often be constants (numbers like 3, 2.9, ½ etc) mixed in as well.</p>
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<p>Never fear! Just multiply the constants separately and put the result in the answer:</p>
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<script type="text/javascript">playerMain("images/variables-exponents-a.txt",600,120);</script>
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<p><i>(Note: "·" means multiply, which we use when the "×" might be confused with the letter "x")</i></p>
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<p>Here is a more complicated example with constants and exponents:</p>
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<script type="text/javascript">playerMain("images/variables-exponents-c.txt",600,120);</script>
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<h2>Negative Exponents</h2>
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<p><b>Negative Exponents Mean Dividing!</b></p>
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<table border="0" align="center">
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<tr>
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<td class="large">x<sup>-1</sup> = <span class="intbl"><em>1</em><strong>x</strong></span></td>
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<td width="60"> </td>
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<td class="large">x<sup>-2</sup> = <span class="intbl"><em>1</em><strong>x<sup>2</sup></strong></span></td>
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<td width="60"> </td>
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<td class="large">x<sup>-3</sup> = <span class="intbl"><em>1</em><strong>x<sup>3</sup></strong></span></td>
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<td width="30"> </td>
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<td>etc...</td>
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</tr>
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</table>
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<p>Get familiar with this idea, it is very important and useful!</p>
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<h2>Dividing</h2>
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<div class="tbl">
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<div class="row"><span class="left">So, how do we do this?</span><span class="right"><span class="intbl"><em>y<sup>3</sup></em><strong>y<sup>2</sup></strong></span></span></div>
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<div class="row"><span class="left">Let's write out all the multiplies:</span><span class="right"><span class="intbl"><em>yyy</em><strong>yy</strong></span></span></div>
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<div class="row">
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<p>Now remove any matching "y"s that are <br>
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both top and bottom (because <span class="large"><span class="intbl"><em>y</em><strong>y</strong></span> = 1</span>)</p>
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</div>
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<div class="row"><span class="left">And we are left with:</span><span class="right">y</span></div>
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</div>
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<p>So 3 "y"s above the line get reduced by 2 "y"s below the line, leaving only 1 "y" :</p>
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<p class="center large"><span class="intbl"><em>y<sup>3</sup></em><strong>y<sup>2</sup></strong></span> = <span class="intbl"><em>yyy</em><strong>yy</strong></span> = y<sup>3-2</sup> = y<sup>1</sup> = y</p>
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<p><b>OR, we could have done it like this:</b></p>
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<p class="center large"><span class="intbl"><em>y<sup>3</sup></em><strong>y<sup>2</sup></strong></span> = y<sup>3</sup>y<sup>-2</sup> = y<sup>3-2</sup> = y<sup>1</sup> = y</p>
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<p>So ... just <b>subtract</b> the exponents of the variables we are dividing by!</p>
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<p>Here is a bigger demonstration, involving several variables:</p>
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<script type="text/javascript">playerMain("images/variables-exponents-d.txt",600,200);</script>
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<p>The "z"s got completely cancelled out! (Which makes sense, because z<sup>2</sup>/z<sup>2</sup> = 1)</p>
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<p>To see what is going on, write down all the multiplies, then "cross out" the variables that are both top and bottom:</p>
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<p class="center large">
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<span class="intbl"><em>x<sup>3</sup> y z<sup>2</sup></em><strong>x y<sup>2</sup> z<sup>2</sup></strong></span>
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=
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<span class="intbl"><em>xxx y zz</em><strong>x yy zz</strong></span>
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=
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<span class="intbl"><em><del>x</del>xx <del>y</del> <del>zz</del></em><strong><del>x</del> <del>y</del>y <del>zz</del></strong></span>
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=
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<span class="intbl"><em>xx</em><strong>y</strong></span>
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=
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<span class="intbl"><em>x<sup>2</sup></em><strong>y</strong></span>
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</p>
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<p>But once again, why <b>count</b> the variables, when the <b>exponents tell you</b> how many?</p>
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<p>Once you get confident you can do the whole thing quite quickly "in place" like this:</p>
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<script type="text/javascript">playerMain("images/variables-exponents-e.txt",600,200);</script>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(325, 326, 2062, 2063, 2314, 2315, 3164, 3165, 3872, 3873);</script> </div>
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<div class="related"> <a href="../exponent.html">Exponent</a> <a href="exponent-fractional.html">Fractional Exponents</a> <a href="exponents-squaring-negative.html">Exponents of Negative Numbers</a> <a href="../index-notation-powers.html">Powers of 10</a> <a href="index.html">Algebra Index</a></div>
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