Solution 1.3:6e

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{{NAVCONTENT_START}}
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Both
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<center> [[Image:1_3_6e.gif]] </center>
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<math>125</math>
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{{NAVCONTENT_STOP}}
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and
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<math>625</math>
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can be written as powers of
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<math>5</math>,
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<math>\begin{align}
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& 125=5\centerdot 5=5\centerdot 5\centerdot 5=5^{3} \\
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& \\
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& 625=5\centerdot 125=5\centerdot 5^{3}=5^{4} \\
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& \\
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\end{align}</math>
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and this means that
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<math>\begin{align}
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& 125^{\frac{1}{2}}=\left( 5^{3} \right)^{\frac{1}{2}}=5^{3\centerdot \frac{1}{2}}=5^{\frac{3}{2}} \\
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& \\
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& 625=\left( 5^{4} \right)^{\frac{1}{3}}=5^{4\centerdot \frac{1}{3}}=5^{\frac{4}{3}} \\
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\end{align}</math>
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From this, we see that
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<math>125^{\frac{1}{2}}>625^{\frac{1}{3}}</math>, since the exponent
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<math>{3}/{2}\;</math>
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is bigger than
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<math>{4}/{3}\;</math>
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and the base
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<math>5</math>
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is bigger than
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<math>1</math>.

Revision as of 12:59, 15 September 2008

Both \displaystyle 125 and \displaystyle 625 can be written as powers of \displaystyle 5,


\displaystyle \begin{align} & 125=5\centerdot 5=5\centerdot 5\centerdot 5=5^{3} \\ & \\ & 625=5\centerdot 125=5\centerdot 5^{3}=5^{4} \\ & \\ \end{align}

and this means that


\displaystyle \begin{align} & 125^{\frac{1}{2}}=\left( 5^{3} \right)^{\frac{1}{2}}=5^{3\centerdot \frac{1}{2}}=5^{\frac{3}{2}} \\ & \\ & 625=\left( 5^{4} \right)^{\frac{1}{3}}=5^{4\centerdot \frac{1}{3}}=5^{\frac{4}{3}} \\ \end{align}


From this, we see that \displaystyle 125^{\frac{1}{2}}>625^{\frac{1}{3}}, since the exponent \displaystyle {3}/{2}\; is bigger than \displaystyle {4}/{3}\; and the base \displaystyle 5 is bigger than \displaystyle 1.

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