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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 data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="vars">
<var id="BASE1" data-ensure="BASE1 !=='n'">randVar()</var>
<var id="BASE2" data-ensure="BASE2 !=='n' && BASE1 !== 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="DEN1">["^", BASE1, EXPDEN1]</var>
<var id="DEN2">["^", BASE2, EXPDEN2]</var>
<var id="DENHINT1">expr(["^", BASE1, EXPDEN1 * EXPDEN3])</var>
<var id="DENHINT2">expr(["^", 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="NUM1">["^", BASE1, EXPNUM1]</var>
<var id="NUM2">["^", BASE2, EXPNUM2]</var>
<var id="NUMHINT1">expr(["^", BASE1, EXPNUM1 * EXPNUM3])</var>
<var id="NUMHINT2">expr(["^", 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">
Rewrite <code>\large{\dfrac{<var>expr(NUM)</var>}{<var>expr(DEN)</var>}}</code> in the form
<code>\large{<var>BASE1</var>^n<var>BASE2</var>^m}</code>.
</p>
<p><em>Assume <code><var>BASE1</var>\neq 0, <var>BASE2</var>\neq 0</code>.</em></p>
<div class="solution" data-type="expression">
<div class="set-sol" data-simplify=""><var>BASE1</var>^<var>EXP1</var> * <var>BASE2</var>^<var>EXP2</var></div>
<div class="set-sol" data-same-form=""><var>BASE1</var>^<var>EXP1</var> * <var>BASE2</var>^<var>EXP2</var></div>
</div>
<div class="hints">
<p>To start, simplify the numerator and the denominator independently.</p>
<div data-if="EXPNUM3 !== 1" data-unwrap="">
<div>
<p>We can use the distributive property of exponents on the numerator.</p>
<p><code>
(\blue{<var>expr(NUM1)</var>}\green{<var>expr(NUM2)</var>})^{<var>EXPNUM3</var>} =
\blue{<var>expr(['^', NUM1, EXPNUM3])</var>}\green{<var>expr(['^', NUM2, EXPNUM3])</var>}
</code></p>
</div>
<div>
<p>
<code>\blue{<var>expr(['^', NUM1, EXPNUM3])</var> = <var>NUMHINT1</var>}</code><br>
<code>\green{<var>expr(['^', NUM2, EXPNUM3])</var> = <var>NUMHINT2</var>}</code>
</p>
<p>So, <code>
(\blue{<var>expr(NUM1)</var>}\green{<var>expr(NUM2)</var>})^{<var>EXPNUM3</var>} =
\blue{<var>NUMHINT1</var>}\green{<var>NUMHINT2</var>}</code>.
</p>
</div>
</div>
<div data-if="EXPDEN3 !== 1" data-unwrap="">
<p>We can use the distributive property of exponents on the denominator.</p>
<p><code>
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>} =
\blue{<var>expr(["^", DEN1, EXPDEN3])</var>}\green{<var>expr(["^", DEN2, EXPDEN3])</var>}
</code></p>
<div>
<p>
<code>\blue{<var>expr(['^', DEN1, EXPDEN3])</var> = <var>DENHINT1</var>}</code><br>
<code>\green{<var>expr(['^', DEN2, EXPDEN3])</var> = <var>DENHINT2</var>}</code>
</p>
<p>So, <code>
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>} =
\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}</code>.
</p>
</div>
</div>
<div>
<p>Therefore, <code>
\dfrac{
<span data-if="EXPNUM3 !== 1">
(\blue{<var>expr(NUM1)</var>}\green{<var>expr(NUM2)</var>})^{<var>EXPNUM3</var>}
</span>
<span data-else="">
\blue{<var>expr(NUM1)</var>}\green{<var>expr(NUM2)</var>}
</span>
}{
<span data-if="EXPDEN3 !== 1">
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>}
</span>
<span data-else="">
\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>}
</span>
} = \dfrac{\blue{<var>NUMHINT1</var>}\green{<var>NUMHINT2</var>}}{
\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}}</code>.
</p>
</div>
<div>
<p>Break up the equation by variable and simplify.</p>
<p><code>
\dfrac{\blue{<var>NUMHINT1</var>}\green{<var>NUMHINT2</var>}}{\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}} =
\blue{\dfrac{<var>NUMHINT1</var>}{<var>DENHINT1</var>}} \cdot \green{\dfrac{<var>NUMHINT2</var>}{<var>DENHINT2</var>}} =
\blue{<var>BASE1</var>^{<var>EXPNUM1 * EXPNUM3</var> - <var>negParens(EXPDEN1 * EXPDEN3)</var>}} \cdot
\green{<var>BASE2</var>^{<var>EXPNUM2 * EXPNUM3</var> - <var>negParens(EXPDEN2 * EXPDEN3)</var>}} =
<var>expr(ANS)</var></code>
</p>
</div>
</div>
</div>
<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="NUM1">["^", BASE1, EXPNUM1]</var>
<var id="NUMHINT1">expr(["^", 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">
Rewrite <code>\large{\dfrac{<var>expr(NUM)</var>}{<var>expr(DEN)</var>}}</code> in the form
<code>\large{<var>BASE1</var>^n<var>BASE2</var>^m}</code>.
</p>
<p><em>Assume <code><var>BASE1</var>\neq 0, <var>BASE2</var>\neq 0</code>.</em></p>
<div class="solution" data-type="expression" data-simplify="">
<var>BASE1</var>^<var>EXP1</var> * <var>BASE2</var>^<var>EXP2</var>
</div>
<div class="hints">
<p>To start, simplify the numerator and the denominator independently.</p>
<div data-if="EXPNUM3 !== 1">
<p>We can use the distributive property of exponents on the numerator.</p>
<p><code>\blue{<var>expr(['^', NUM1, EXPNUM3])</var>} = \blue{<var>NUMHINT1</var>}</code></p>
</div>
<div data-if="EXPDEN3 !== 1" data-unwrap="">
<p>We can use the distributive property of exponents on the denominator.</p>
<p><code>
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>} =
\blue{<var>expr(["^", DEN1, EXPDEN3])</var>}\green{<var>expr(["^", DEN2, EXPDEN3])</var>}
</code></p>
<div>
<p>
<code>\blue{<var>expr(['^', DEN1, EXPDEN3])</var> = <var>DENHINT1</var>}</code><br>
<code>\green{<var>expr(['^', DEN2, EXPDEN3])</var> = <var>DENHINT2</var>}</code>
</p>
<p>So, <code>
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>} =
\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}</code>.
</p>
</div>
</div>
<div>
<p>Therefore, <code>
\dfrac{
<span data-if="EXPNUM3 !== 1">(\blue{<var>expr(NUM1)</var>})^{<var>EXPNUM3</var>}</span>
<span data-else="">\blue{<var>expr(NUM1)</var>}</span>
}{
<span data-if="EXPDEN3 !== 1">
(\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>})^{<var>EXPDEN3</var>}
</span>
<span data-else="">
\blue{<var>expr(DEN1)</var>}\green{<var>expr(DEN2)</var>}
</span>
} = \dfrac{\blue{<var>NUMHINT1</var>}}{
\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}}</code>.
</p>
</div>
<div>
<p>Break up the equation by variable and simplify.</p>
<p><code>
\dfrac{\blue{<var>NUMHINT1</var>}}{\blue{<var>DENHINT1</var>}\green{<var>DENHINT2</var>}} =
\blue{\dfrac{<var>NUMHINT1</var>}{<var>DENHINT1</var>}} \cdot \green{\dfrac{1}{<var>DENHINT2</var>}} =
\blue{<var>BASE1</var>^{<var>EXPNUM1 * EXPNUM3</var> - <var>negParens(EXPDEN1 * EXPDEN3)</var>}} \cdot
\green{<var>BASE2</var>^{ - <var>negParens(EXPDEN2 * EXPDEN3)</var>}} =
<var>expr(ANS)</var></code>
</p>
</div>
</div>
</div>
</div>
</div>
</body>
</html>
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