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. \(a^{-n} = \frac{1}{a^n}\).

Learn how to simplify the expression (2a^(-2)*b)^(-1)*(b^2/a)^2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including exponents, fractional exponents, and simplifying expressions. Suitable for high school students.

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