Solution 1.3:6e

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Current revision (14:57, 22 September 2008) (edit) (undo)
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Both
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Both 125 and 625 can be written as powers of 5,
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<math>125</math>
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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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{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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125 &= 5\cdot 5 = 5\cdot 5\cdot 5 = 5^{3},\\[5pt]
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& 125=5\centerdot 5=5\centerdot 5\centerdot 5=5^{3} \\
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625 &= 5\cdot 125 = 5\cdot 5^{3} = 5^{4},
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& \\
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\end{align}</math>}}
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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
and this means that
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{{Displayed math||<math>\begin{align}
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125^{\frac{1}{2}} &= \bigl(5^{3}\bigr)^{\frac{1}{2}} = 5^{3\cdot\frac{1}{2}} = 5^{\frac{3}{2}},\\[5pt]
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625 &= \bigl(5^{4}\bigr)^{\frac{1}{3}} = 5^{4\cdot\frac{1}{3}} = 5^{\frac{4}{3}}\,\textrm{.}
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\end{align}</math>}}
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<math>\begin{align}
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From this, we see that <math>125^{\frac{1}{2}} > 625^{\frac{1}{3}}</math>, since the exponent 3/2 is bigger than 4/3 and the base 5 is bigger than 1.
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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>.
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Current revision

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

\displaystyle \begin{align}

125 &= 5\cdot 5 = 5\cdot 5\cdot 5 = 5^{3},\\[5pt] 625 &= 5\cdot 125 = 5\cdot 5^{3} = 5^{4}, \end{align}

and this means that

\displaystyle \begin{align}

125^{\frac{1}{2}} &= \bigl(5^{3}\bigr)^{\frac{1}{2}} = 5^{3\cdot\frac{1}{2}} = 5^{\frac{3}{2}},\\[5pt] 625 &= \bigl(5^{4}\bigr)^{\frac{1}{3}} = 5^{4\cdot\frac{1}{3}} = 5^{\frac{4}{3}}\,\textrm{.} \end{align}

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

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