# How do you simplify 1000^(-2/3)?

Jul 30, 2017

See a solution process below:

#### Explanation:

First, we can use this rule for exponents to eliminate the negative exponent:

${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$

${1000}^{\textcolor{red}{- \frac{2}{3}}} = \frac{1}{1000} ^ \textcolor{red}{- - \frac{2}{3}} = \frac{1}{1000} ^ \textcolor{red}{\frac{2}{3}}$

We can next rewrite the exponent as:

$\frac{1}{1000} ^ \textcolor{red}{\frac{2}{3}} = \frac{1}{1000} ^ \left(\frac{1}{3} \times 2\right)$

And then use this rule to rewrite the expression again:

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

$\frac{1}{1000} ^ \left(\textcolor{red}{\frac{1}{3}} \times \textcolor{b l u e}{2}\right) = \frac{1}{{1000}^{\textcolor{red}{\frac{1}{3}}}} ^ \textcolor{b l u e}{2}$

We can convert the term within the parenthesis to radical form using this rule:

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

$\frac{1}{{1000}^{\textcolor{red}{\frac{1}{3}}}} ^ \textcolor{b l u e}{2} = \frac{1}{\sqrt[\textcolor{red}{3}]{1000}} ^ \textcolor{b l u e}{2} = \frac{1}{10} ^ \textcolor{b l u e}{2} = \frac{1}{100}$