Answer
$\frac{1}{a^{\frac{9}{2}}}$
Work Step by Step
We are given the expression $(a^{\frac{1}{2}}a^{-2})^{3}$.
First, we can use the product rule to simplify, which holds that $a^{m}\times a^{n}=a^{m+n}$ (where a is a real number, and m and n are positive integers).
$(a^{\frac{1}{2}+(-2)})^{3}=(a^{-\frac{3}{2}})^{3}$
Next, we can use the power rule, which holds that $(a^{m})^{n}=a^{m\times n}$ (where a is a real number, and m and n are integers).
$(a^{-\frac{3}{2}})^{3}=a^{-\frac{3}{2}\times3}=a^{-\frac{9}{2}}=\frac{1}{a^{\frac{9}{2}}}$
