Solution 1.3:6d

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Current revision (14:52, 22 September 2008) (edit) (undo)
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One way to compare the two numbers is to rewrite the power
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One way to compare the two numbers is to rewrite the power <math>\bigl(5^{\frac{1}{3}}\bigr)^{4}</math> so that it has the same exponent as <math>400^{\frac{1}{3}}</math>,
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<math>\left( 5^{\frac{1}{3}} \right)^{4}</math>
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so that it has the same exponent as
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<math>400^{\frac{1}{3}}</math>,
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{{Displayed math||<math>\bigl(5^{\frac{1}{3}}\bigr)^{4} = 5^{\frac{1}{3}\cdot 4} = 5^{4\cdot\frac{1}{3}} = \bigl(5^{4}\bigr)^{\frac{1}{3}} = \bigl(5\cdot 5\cdot 5\cdot 5\bigr)^{\frac{1}{3}} = 625^{\frac{1}{3}}\,</math>.}}
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<math>\left( 5^{\frac{1}{3}} \right)^{4}=5^{\frac{1}{3}\centerdot 4}=5^{4\centerdot \frac{1}{3}}=\left( 5^{4} \right)^{\frac{1}{3}}=\left( 5\centerdot 5\centerdot 5\centerdot 5 \right)^{\frac{1}{3}}=625^{\frac{1}{3}}</math>.
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Now, we see that <math>\bigl(5^{\frac{1}{3}}\bigr)^{4} > 400^{\frac{1}{3}}</math>, because <math>625 > 400</math> and the exponent 1/3 is positive.
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Now, we see that
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<math>\left( 5^{\frac{1}{3}} \right)^{4}>400^{\frac{1}{3}}</math>, because
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<math>625>400</math>
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and the exponent
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<math>\frac{1}{3}</math>
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is positive.
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Current revision

One way to compare the two numbers is to rewrite the power \displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} so that it has the same exponent as \displaystyle 400^{\frac{1}{3}},

\displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} = 5^{\frac{1}{3}\cdot 4} = 5^{4\cdot\frac{1}{3}} = \bigl(5^{4}\bigr)^{\frac{1}{3}} = \bigl(5\cdot 5\cdot 5\cdot 5\bigr)^{\frac{1}{3}} = 625^{\frac{1}{3}}\,.

Now, we see that \displaystyle \bigl(5^{\frac{1}{3}}\bigr)^{4} > 400^{\frac{1}{3}}, because \displaystyle 625 > 400 and the exponent 1/3 is positive.

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