Answer
$\frac{8a^{\frac{3}{4}}}{27}$
Work Step by Step
First, we will simplify the expression within the parenthesis using the rule $\frac{a^{m}}{a^{n}}=a^{m-n}$:
$(\frac{2a^{\frac{1}{2}}}{3a^{\frac{1}{4}}})^{3}$
=$(\frac{2}{3}\times\frac{a^{\frac{1}{2}}}{a^{\frac{1}{4}}})^{3}$
=$(\frac{2}{3}\times a^{\frac{1}{2}-\frac{1}{4}})^{3}$
=$(\frac{2}{3}\times a^{\frac{2-1}{4}})^{3}$
=$(\frac{2}{3}\times a^{\frac{1}{4}})^{3}$
Now, we raise each term in the expression to the power of $3$:
$(\frac{2}{3}\times a^{\frac{1}{4}})^{3}$
=$(\frac{2}{3})^{3}\times (a^{\frac{1}{4}})^{3}$
=$(\frac{2^{3}}{3^{3}})\times (a^{\frac{3}{4}})$
=$(\frac{8}{27})\times (a^{\frac{3}{4}})$
=$\frac{8a^{\frac{3}{4}}}{27}$
