Simplify the expression $\left( \frac { 9 m ^ { \frac { 1 } { 6 } } n ^ { \frac { 1 } { 3 } } } { 4 n ^ { \frac { 2 } { 3 } } } \right) ^ { 2 }$ .
A) $\frac { 9 m ^ { \frac { 1 } { 6 } } } { 4 n ^ { 2 } }$
B) $\frac { 81 m ^ { \frac { 1 } { 3 } } } { 16 n ^ { \frac { 2 } { 3 } } }$
C) $\frac { 81 m ^ { \frac { 1 } { 5 } } n ^ { \frac { 2 } { 3 } } } { 16 }$
D) $\frac { 9 m ^ { \frac { 1 } { 2 } } } { 4 n ^ { \frac { 2 } { 3 } } }$
E) $\frac { 81 m ^ { \frac { 1 } { 4 } } } { 16 n ^ { 2 } }$

Question

Quizplus · Accepted Answer

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 $\frac{81m^{\frac{1}{3}}}{16n^{\frac{2}{3}}}$.

Simplify the Expression (9m16n134n23)2\left( \frac { 9 m ^ { \frac { 1 } { 6 } } n ^ { \frac { 1 } { 3 } } } { 4 n ^ { \frac { 2 } { 3 } } } \right) ^ { 2 }(4n32​9m61​n31​​)2

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