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<!DOCTYPE html>
<html data-require="math math-format expressions">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Simplifying expressions with exponents</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars" data-ensure="BASE1 !== BASE2">
<var id="BASE1">randVar( )</var>
<var id="BASE2">randVar( )</var>
<var id="EXPNUM1">randRangeNonZero( -5, 5 )</var>
<var id="EXPNUM3">randRangeNonZero( -5, 5 )</var>
<var id="EXPDEN1">randRangeNonZero( -5, 5 )</var>
<var id="EXPDEN2">randRangeNonZero( -5, 5 )</var>
<var id="EXPDEN3">randRangeNonZero( -5, 5 )</var>
<var id="DEN">[ "^", [ "*", [ "^", BASE1, EXPDEN1 ], [ "^", BASE2, EXPDEN2 ] ], EXPDEN3 ]</var>
<var id="DENHINT1">[ "*", [ "^", [ "^", BASE1, EXPDEN1 ], EXPDEN3 ], [ "^", [ "^", BASE2, EXPDEN2 ], EXPDEN3 ] ]</var>
<var id="DENHINT2">[ "*", [ "^", BASE1, EXPDEN1 * EXPDEN3 ], [ "^", BASE2, EXPDEN2 * EXPDEN3 ] ]</var>
</div>
<div class="problems">
<div id="simplify-exponential-form-full" data-weight="2">
<div class="vars">
<var id="EXPNUM2">randRangeNonZero( -5, 5 )</var>
<var id="NUM">[ "^", [ "*", [ "^", BASE1, EXPNUM1 ], [ "^", BASE2, EXPNUM2 ] ], EXPNUM3 ]</var>
<var id="NUMHINT1">[ "*", [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ], [ "^", [ "^", BASE2, EXPNUM2 ], EXPNUM3 ] ]</var>
<var id="NUMHINT2">[ "*", [ "^", BASE1, EXPNUM1 * EXPNUM3 ], [ "^", BASE2, EXPNUM2 * EXPNUM3 ] ]</var>
<var id="EXP1">EXPNUM1 * EXPNUM3 - EXPDEN1 * EXPDEN3</var>
<var id="EXP2">EXPNUM2 * EXPNUM3 - EXPDEN2 * EXPDEN3</var>
<var id="ANS">[ "*", [ "^", BASE1, EXP1 ], [ "^", BASE2, EXP2 ] ]</var>
</div>
<p class="question">Simplify; express your answer in exponential form. Assume <code><var>BASE1</var>\neq 0, <var>BASE2</var>\neq 0</code>.</p>
<p><code>\dfrac{\color{orange}{<var>expr( NUM )</var>}}{\color{green}{<var>expr( DEN )</var>}}</code></p>
<div class="solution" data-type="multiple">
<p><code><var>BASE1</var></code><span class="sol exp short30" data-forms="integer"><var>EXP1</var></span><code><var>BASE2</var></code><span class="sol exp short30" data-forms="integer"><var>EXP2</var></span></p>
<p class="example">enter a (possibly negative) integer for each exponent</p>
</div>
<div class="hints">
<p>To start, try simplifying the numerator and the denominator independently.</p>
<div>
<p>In the numerator, we can use the distributive property of exponents.</p>
<p><code>\color{orange}{<var>expr( NUM )</var> = <var>expr( NUMHINT1 )</var>}</code>.</p>
</div>
<p>On the left, we have <code>\color{orange}{<var>expr( [ "^", BASE1, EXPNUM1 ] )</var>}</code> to the exponent <code>\color{orange}{<var>EXPNUM3</var>}</code>. Now <code>\color{orange}{<var>EXPNUM1</var> \times <var>EXPNUM3</var> = <var>EXPNUM1 * EXPNUM3</var>}</code>, so <code>\color{orange}{<var>expr( [ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ] )</var> = <var>expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )</var>}</code>.</p>
<div>
<p>Apply the ideas above to simplify the equation.<p>
<p><code>\dfrac{\color{orange}{<var>expr( NUM )</var>}}{\color{green}{<var>expr( DEN )</var>}} = \dfrac{\color{orange}{<var>expr( NUMHINT2 )</var>}}{\color{green}{<var>expr( DENHINT2 )</var>}}</code>.</p>
</div>
<div>
<p>Break up the equation by variable and simplify.</p>
<p><code>\dfrac{\color{orange}{<var>expr( NUMHINT2 )</var>}}{\color{green}{<var>expr( DENHINT2 )</var>}} = \dfrac{\color{orange}{<var>expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )</var>}}{\color{green}{<var>expr( [ "^", BASE1, EXPDEN1 * EXPDEN3 ] )</var>}} \cdot \dfrac{\color{orange}{<var>expr( [ "^", BASE2, EXPNUM2 * EXPNUM3 ] )</var>}}{\color{green}{<var>expr( [ "^", BASE2, EXPDEN2 * EXPDEN3 ] )</var>}} = <var>BASE1</var>^{\color{orange}{<var>EXPNUM1 * EXPNUM3</var>} - \color{green}{<var>negParens( EXPDEN1 * EXPDEN3 )</var>}} \cdot <var>BASE2</var>^{\color{orange}{<var>EXPNUM2 * EXPNUM3</var>} - \color{green}{<var>negParens( EXPDEN2 * EXPDEN3 )</var>}} = <var>expr( ANS )</var></code></p>
</div>
</div>
</div> <!-- simplify-exponential-form-full-->
<div id="simplify-exponential-form-mini" data-weight="3">
<div class="vars">
<var id="EXPNUM2">0</var>
<var id="NUM">[ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ]</var>
<var id="NUMHINT1">[ "^", [ "^", BASE1, EXPNUM1 ], EXPNUM3 ]</var>
<var id="NUMHINT2">[ "^", BASE1, EXPNUM1 * EXPNUM3 ]</var>
<var id="EXP1">EXPNUM1 * EXPNUM3 - EXPDEN1 * EXPDEN3</var>
<var id="EXP2">EXPNUM2 * EXPNUM3 - EXPDEN2 * EXPDEN3</var>
<var id="ANS">[ "*", [ "^", BASE1, EXP1 ], [ "^", BASE2, EXP2 ] ]</var>
</div>
<p class="question">Simplify; express your answer in exponential form. Assume <code><var>BASE1</var>\neq 0, <var>BASE2</var>\neq 0</code>.</p>
<p><code>\dfrac{\color{orange}{<var>expr( NUM )</var>}}{\color{green}{<var>expr( DEN )</var>}}</code></p>
<div class="solution" data-type="multiple">
<p><code><var>BASE1</var></code><span class="sol exp short30" data-forms="integer"><var>EXP1</var></span><code><var>BASE2</var></code><span class="sol exp short30" data-forms="integer"><var>EXP2</var></span></p>
<p class="example">enter a (possibly negative) integer for each exponent</p>
</div>
<div class="hints">
<p>To start, try working on the numerator and the denominator independently.</p>
<p>In the numerator, we have <code>\color{orange}{<var>expr( [ "^", BASE1, EXPNUM1 ] )</var>}</code> to the exponent <code>\color{orange}{<var>EXPNUM3</var>}</code>. Now <code>\color{orange}{<var>EXPNUM1</var> \times <var>EXPNUM3</var> = <var>EXPNUM1 * EXPNUM3</var>}</code>, so <code>\color{orange}{<var>expr( NUM )</var> = <var>expr( NUMHINT2 )</var>}</code>.</p>
<div>
<p>In the denominator, we can use the distributive property of exponents.</p>
<p><code>\color{green}{<var>expr( DEN )</var> = <var>expr( DENHINT1 )</var>}</code>.</p>
</div>
<div>
<p>Simplify using the same method from the numerator and put the entire equation together.</p>
<p><code>\dfrac{\color{orange}{<var>expr( NUM )</var>}}{\color{green}{<var>expr( DEN )</var>}} = \dfrac{\color{orange}{<var>expr( NUMHINT2 )</var>}}{\color{green}{<var>expr( DENHINT2 )</var>}}</code>.</p>
</div>
<div>
<p>Break up the equation by variable and simplify.</p>
<p><code>\dfrac{\color{orange}{<var>expr( NUMHINT2 )</var>}}{\color{green}{<var>expr( DENHINT2 )</var>}} = \dfrac{\color{orange}{<var>expr( [ "^", BASE1, EXPNUM1 * EXPNUM3 ] )</var>}}{\color{green}{<var>expr( [ "^", BASE1, EXPDEN1 * EXPDEN3 ] )</var>}} \cdot \dfrac{\color{orange}{1}}{\color{green}{<var>expr( [ "^", BASE2, EXPDEN2 * EXPDEN3 ] )</var>}} = <var>BASE1</var>^{\color{orange}{<var>EXPNUM1 * EXPNUM3</var>} - \color{green}{<var>negParens( EXPDEN1 * EXPDEN3 )</var>}} \cdot <var>BASE2</var>^{- \color{green}{<var>negParens( EXPDEN2 * EXPDEN3 )</var>}} = <var>expr( ANS )</var></code>.</p>
</div>
</div>
</div> <!-- simplify-exponential-form-mini-->
</div> <!-- problems -->
</div> <!-- exercise -->
</body>
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