How do you simplify ((3m^(1/6) n^(1/3)) / (4n^(-2/3))) ^2?

Feb 12, 2017

See the entire simplification process below:

Explanation:

First, we will use this rule of exponents to simplify the term within the parenthesis:

${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$

${\left(\frac{3 {m}^{\frac{1}{6}} {n}^{\textcolor{red}{\frac{1}{3}}}}{4 {n}^{\textcolor{b l u e}{- \frac{2}{3}}}}\right)}^{2} = {\left(\frac{3 {m}^{\frac{1}{6}} {n}^{\textcolor{red}{\frac{1}{3}} - \textcolor{b l u e}{- \frac{2}{3}}}}{4}\right)}^{2} = {\left(\frac{3 {m}^{\frac{1}{6}} {n}^{\frac{3}{3}}}{4}\right)}^{2} =$

${\left(\frac{3 {m}^{\frac{1}{6}} {n}^{1}}{4}\right)}^{2}$

Now, use these two rules for exponents to simplify the terms for the entire expression:

$a = {a}^{\textcolor{red}{1}}$ and ${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left(\frac{{3}^{\textcolor{red}{1}} {m}^{\textcolor{red}{\frac{1}{6}}} {n}^{\textcolor{red}{1}}}{4} ^ \textcolor{red}{1}\right)}^{\textcolor{b l u e}{2}} = \frac{{3}^{\textcolor{red}{1} \times \textcolor{b l u e}{2}} {m}^{\textcolor{red}{\frac{1}{6}} \times \textcolor{b l u e}{2}} {n}^{\textcolor{red}{1} \times \textcolor{b l u e}{2}}}{4} ^ \left(\textcolor{red}{1} \times \textcolor{b l u e}{2}\right) = \frac{{3}^{2} {m}^{\frac{2}{6}} {n}^{2}}{4} ^ 2 = \frac{9 {m}^{\frac{1}{3}} {n}^{2}}{16}$