Mathematics, 05.07.2019 09:00 wiltseliz4800
Hello is there a way to calculate 0.5^0.6 without using a calculator cause i am not allowed to use one during my exam. in advance
Answers: 1
![\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 0.\underline{5}\implies \cfrac{05}{1\underline{0}}\implies \stackrel{simplified}{\cfrac{1}{2}}~\hfill 0.\underline{6}\implies \cfrac{06}{1\underline{0}}\implies \stackrel{simplfied}{\cfrac{3}{5}}](/tpl/images/0053/5116/7ffdc.png)
![\bf ~\dotfill\\\\ 0.5^{0.6}\implies \left( \cfrac{1}{2} \right)^{\frac{3}{5}}\implies \sqrt[5]{\left( \cfrac{1}{2} \right)^3}\implies \sqrt[5]{\cfrac{1^3}{2^3}}\implies \cfrac{\sqrt[5]{1^3}}{\sqrt[5]{2^3}}\implies \cfrac{\sqrt[5]{1}}{\sqrt[5]{8}} \\\\\\ \textit{and since }1^5=1\qquad \cfrac{\sqrt[5]{1^5}}{\sqrt[5]{8}}\implies \cfrac{1}{\sqrt[5]{8}}~\hfill \stackrel{\textit{and if we rationalize the denominator}}{\cfrac{\sqrt[5]{4}}{2}}](/tpl/images/0053/5116/15f24.png)
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