How do you write ((5^(2/3))^(1/5))^2 in radical form?

Jan 29, 2018

See a solution process below:

Explanation:

First, use this rule of exponents three times to simplify the expression:

${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left({\left({5}^{\frac{2}{3}}\right)}^{\textcolor{red}{\frac{1}{5}}}\right)}^{\textcolor{b l u e}{2}} \implies {\left({5}^{\frac{2}{3}}\right)}^{\textcolor{red}{\frac{1}{5}} \times \textcolor{b l u e}{2}} \implies {\left({5}^{\frac{2}{3}}\right)}^{\textcolor{b l u e}{\frac{2}{5}}}$

${\left({5}^{\textcolor{red}{\frac{2}{3}}}\right)}^{\textcolor{b l u e}{\frac{2}{5}}} \implies {5}^{\textcolor{red}{\frac{2}{3}} \times \textcolor{b l u e}{\frac{2}{5}}} \implies {5}^{\frac{4}{15}}$

Next, use the same rule in reverse to rewrite the expression as:

${5}^{\frac{4}{15}} \implies {5}^{\textcolor{red}{4} \times \textcolor{b l u e}{\frac{1}{15}}} \implies {\left({5}^{\textcolor{red}{4}}\right)}^{\textcolor{b l u e}{\frac{1}{15}}} \implies {625}^{\frac{1}{15}}$

Now, use this rule of radicals and exponents to rewrite the expression in radical form:

${x}^{\frac{1}{\textcolor{red}{n}}} = \sqrt[\textcolor{red}{n}]{x}$

${625}^{\frac{1}{\textcolor{red}{15}}} = \sqrt[\textcolor{red}{15}]{625}$