Simplify the expression ${\left(\frac{9m^{\frac{1}{6}}{n}^{\frac{1}{3}}}{4n^{\frac{2}{3}}}\right)}^2$ .

A) $\frac{9m^{\frac{1}{6}}}{4n^2}$
B) $\frac{81m^{\frac{1}{3}}}{16n^{\frac{2}{3}}}$
C) $\frac{81m^{\frac{1}{5}}{n}^{\frac{2}{3}}}{16}$
D) $\frac{9m^{\frac{1}{2}}}{4n^{\frac{2}{3}}}$
E) $\frac{81m^{\frac{1}{4}}}{16n^2}$

First, simplify the expression inside the parentheses before applying the exponent of 2. The exponent on $n$ in the denominator simplifies with the exponent on $n$ in the numerator: $n^{\frac{1}{3}}/n^{\frac{2}{3}}=n^{-\frac{1}{3}}$ . When you square the entire expression, you square both the numerical coefficient (9/4)^2 = 81/16 and the exponents on $m$ and $n$ . The exponent on $m$ becomes $2\times \frac{1}{6}=\frac{1}{3}$ , and the exponent on $n$ becomes $2\times -\frac{1}{3}=-\frac{2}{3}$ , resulting in $\frac{81m^{\frac{1}{3}}}{16n^{\frac{2}{3}}}$ .