Question

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. Thanks in advance

Answer 1

recall that we can change any decimal number to a fraction by simply using as many zeros at the bottom as there are decimals, and lose the dot atop, so let's do so, this one has 1 decimal each, so that means one zero at the bottom of the fractions.

`\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{( n)/( m)} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-( n)/( m)} \implies \cfrac{1}{a^{( 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}}`

`\bf ~\dotfill\\\\ 0.5^(0.6)\implies \left( \cfrac{1}{2} \right)^{(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}}`