How do you find the value of 125^(-2/3)?

Apr 11, 2017

See the entire solution process below:

Explanation:

First, we can rewrite this expression using this rule of exponents:

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

${125}^{- \frac{2}{3}} = {125}^{\textcolor{red}{\frac{1}{3}} \times \textcolor{b l u e}{- 2}} = {\left({125}^{\textcolor{red}{\frac{1}{3}}}\right)}^{\textcolor{b l u e}{- 2}}$

We can next rewrite the term inside the parenthesis using this rule for exponents and radicals:

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

${\left({125}^{\textcolor{red}{\frac{1}{\textcolor{red}{3}}}}\right)}^{\textcolor{b l u e}{- 2}} = {\left(\sqrt[\textcolor{red}{3}]{125}\right)}^{\textcolor{b l u e}{- 2}} = {5}^{\textcolor{b l u e}{- 2}}$

Now, use this rule of exponents to complete the evaluation:

${x}^{\textcolor{b l u e}{a}} = \frac{1}{x} ^ \textcolor{b l u e}{- a}$

${5}^{\textcolor{b l u e}{- 2}} = \frac{1}{5} ^ \textcolor{b l u e}{- - 2} = \frac{1}{5} ^ 2 = \frac{1}{25} = 0.04$