We are asked to simplify the expression $(a^{-2}b^3)^{-2}$, writing the answer with positive powers.

1. Problem Description

We are asked to simplify the expression $(a^{-2}b^3)^{-2}$, writing the answer with positive powers.

2. Solution Steps

We will use the power of a product rule and the power of a power rule to simplify the expression. The power of a product rule states that $(xy)^n = x^n y^n$. The power of a power rule states that $(x^m)^n = x^{mn}$.

$(a^{-2}b^3)^{-2} = (a^{-2})^{-2}(b^3)^{-2}$

$= a^{(-2)(-2)}b^{(3)(-2)}$

$= a^4 b^{-6}$

Since we need to write the answer with positive powers, we rewrite $b^{-6}$ as $\frac{1}{b^6}$.

$a^4 b^{-6} = a^4 \cdot \frac{1}{b^6} = \frac{a^4}{b^6}$

3. Final Answer

$\frac{a^4}{b^6}$