Lösung 1.3:6e
Aus Online Mathematik Brückenkurs 1
(Unterschied zwischen Versionen)
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Both 125 and 625 can be written as powers of 5, | Both 125 and 625 can be written as powers of 5, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>\begin{align} |
125 &= 5\cdot 5 = 5\cdot 5\cdot 5 = 5^{3},\\[5pt] | 125 &= 5\cdot 5 = 5\cdot 5\cdot 5 = 5^{3},\\[5pt] | ||
625 &= 5\cdot 125 = 5\cdot 5^{3} = 5^{4}, | 625 &= 5\cdot 125 = 5\cdot 5^{3} = 5^{4}, | ||
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and this means that | and this means that | ||
| - | {{ | + | {{Abgesetzte Formel||<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] | 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{.} | 625 &= \bigl(5^{4}\bigr)^{\frac{1}{3}} = 5^{4\cdot\frac{1}{3}} = 5^{\frac{4}{3}}\,\textrm{.} | ||
Version vom 08:19, 22. Okt. 2008
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.
