Solution 1.3:6e
From Förberedande kurs i matematik 1
(Difference between revisions)
m (Lösning 1.3:6e moved to Solution 1.3:6e: Robot: moved page) |
m |
||
| (One intermediate revision not shown.) | |||
| Line 1: | Line 1: | ||
| - | {{ | + | 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.
