# How do you evaluate and simplify (120^(-2/5)*120^(2/5))/7^(-3/4)?

Jul 27, 2017

See a solution process below:

#### Explanation:

First, use these rules of exponents to simplify the numerator:

${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$ and ${a}^{\textcolor{red}{0}} = 1$

$\frac{{120}^{\textcolor{red}{- \frac{2}{5}}} \cdot {120}^{\textcolor{b l u e}{\frac{2}{5}}}}{7} ^ \left(- \frac{3}{4}\right) \implies {120}^{\textcolor{red}{- \frac{2}{5}} + \textcolor{b l u e}{\frac{2}{5}}} / {7}^{- \frac{3}{4}} \implies$

${120}^{\textcolor{red}{0}} / {7}^{- \frac{3}{4}} \implies \frac{1}{7} ^ \left(- \frac{3}{4}\right)$

Next, we will use this rule to rewrite the expression:

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

$\frac{1}{7} ^ \textcolor{red}{- \frac{3}{4}} = {7}^{\textcolor{red}{- - \frac{3}{4}}} = {7}^{\frac{3}{4}}$

Then, we can rewrite the expression as:

${7}^{3 \times \frac{1}{4}}$

Now, we can use this rule of exponents to continue the simplification:

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

${7}^{\textcolor{red}{3} \times \textcolor{b l u e}{\frac{1}{4}}} \implies {\left({7}^{\textcolor{red}{3}}\right)}^{\textcolor{b l u e}{\frac{1}{4}}} \implies {343}^{\frac{1}{4}}$

Or, using this rule: ${x}^{\frac{1}{\textcolor{red}{n}}} = \sqrt[\textcolor{red}{n}]{x}$

$\sqrt{343}$