To simplify the expression $(2a^{-2}b)^{-1} \cdot \left(\frac{b^2}{a}\right)^2$, let's break it down step by step.

Step 1: Simplify each part of the expression.

Part 1: Simplify $(2a^{-2}b)^{-1}$

• The expression inside the parentheses is $2a^{-2}b$.
• The reciprocal (indicated by the exponent $-1$) of $2a^{-2}b$ is: $\frac{1}{2a^{-2}b} = \frac{1}{2} \cdot a^2 \cdot \frac{1}{b} = \frac{a^2}{2b}$

Part 2: Simplify $\left(\frac{b^2}{a}\right)^2$

• Square each part of the fraction: $\left(\frac{b^2}{a}\right)^2 = \frac{(b^2)^2}{a^2} = \frac{b^4}{a^2}$

Step 2: Multiply the simplified parts together.

Now, multiply the simplified parts: $\frac{a^2}{2b} \cdot \frac{b^4}{a^2}$

Step 3: Simplify the resulting expression.

• Multiply the numerators together and the denominators together: $\frac{a^2 \cdot b^4}{2b \cdot a^2}$
• Notice that $a^2$ in the numerator and denominator cancels out: $\frac{b^4}{2b} = \frac{b^{4-1}}{2} = \frac{b^3}{2}$

Final Answer:

The simplified expression is: $\boxed{\frac{b^3}{2}}$

Would you like more details or have any questions about this solution?

Here are five related questions you might find useful:

1. How would the expression change if the exponent $-1$ applied only to $a^{-2}$?
2. How do you simplify $(a^{-3}b^2)^{-2} \cdot (ab^{-1})^3$?
3. What are the general rules for simplifying expressions with negative exponents?
4. How do you simplify expressions involving fractional exponents?
5. What is the result of $\left(\frac{2a^{-1}}{3b^{-2}}\right)^3$?

Tip: When working with negative exponents, remember that they indicate reciprocals. $a^{-n} = \frac{1}{a^n}$.