Simplify the expression $\sqrt[3]{a^2} \times \sqrt{a^{-3}}$.

1. Problem Description

Simplify the expression $\sqrt[3]{a^2} \times \sqrt{a^{-3}}$.

2. Solution Steps

First, rewrite the radicals using fractional exponents.

$\sqrt[3]{a^2} = a^{\frac{2}{3}}$

$\sqrt{a^{-3}} = a^{-\frac{3}{2}}$

Now, we have $a^{\frac{2}{3}} \times a^{-\frac{3}{2}}$.

Using the rule $a^m \times a^n = a^{m+n}$, we have

$a^{\frac{2}{3} + (-\frac{3}{2})} = a^{\frac{2}{3} - \frac{3}{2}}$.

To add the fractions, find a common denominator, which is

6. $\frac{2}{3} - \frac{3}{2} = \frac{2 \times 2}{3 \times 2} - \frac{3 \times 3}{2 \times 3} = \frac{4}{6} - \frac{9}{6} = \frac{4-9}{6} = -\frac{5}{6}$

Therefore, the expression simplifies to $a^{-\frac{5}{6}}$.

3. Final Answer

$a^{-\frac{5}{6}}$