Solution 1.3:6d

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One way to compare the two numbers is to rewrite the power
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<center> [[Image:1_3_6d.gif]] </center>
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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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<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
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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.

Revision as of 12:56, 15 September 2008

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


\displaystyle \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}}.

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

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