Solution 1.3:6e
From Förberedande kurs i matematik 1
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- | {{ | + | Both 125 and 625 can be written as powers of 5, |
- | < | + | |
- | {{ | + | {{Displayed math||<math>\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}</math>}} | ||
+ | |||
+ | and this means that | ||
+ | |||
+ | {{Displayed math||<math>\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}</math>}} | ||
+ | |||
+ | 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. |
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.