# Evaluate the following?

Oct 26, 2016

${x}^{\frac{77}{15}} = \sqrt[15]{{x}^{77}}$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of exponents}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{a}^{\frac{m}{n}} = \sqrt[n]{{a}^{m}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

and $\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{a}^{m} \times {a}^{n} = {a}^{m + n} , {a}^{m} / {a}^{n} = {a}^{m - n} , {a}^{-} m \Leftrightarrow \frac{1}{a} ^ m} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

The above expression can be written as.

${x}^{\frac{3}{2}} \times {x}^{\frac{5}{3}} \times {x}^{\frac{77}{30}} \times {x}^{- \frac{3}{5}}$

$= {x}^{\frac{3}{2} + \frac{5}{3} + \frac{77}{30} - \frac{3}{5}}$

This now becomes a fraction exercise. That is.

$\frac{3}{2} + \frac{5}{3} + \frac{77}{30} - \frac{3}{5}$

$= \frac{45}{30} + \frac{50}{30} + \frac{77}{30} - \frac{18}{30} = \frac{154}{30} = \frac{77}{15}$

Thus our simplification is.

x^(77/15)=root(15)(x^(77)