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The expression you provided is:

\[
(5w^2 z^{-2})^2 (z^3)
\]

Let me simplify this step by step:

1. First, simplify \((5w^2 z^{-2})^2\):
   - The \(5\) gets squared: \(5^2 = 25\).
   - For \(w^2\), the exponent gets multiplied by 2: \((w^2)^2 = w^{4}\).
   - For \(z^{-2}\), the exponent gets multiplied by 2: \((z^{-2})^2 = z^{-4}\).

So, \((5w^2 z^{-2})^2 = 25w^4 z^{-4}\).

2. Now, multiply the result by \(z^3\):
   \[
   25w^4 z^{-4} \cdot z^3 = 25w^4 z^{-1}
   \]

Thus, the simplified expression is:

\[
25w^4 z^{-1}
\]

Let me know if you'd like further details on any step, or if you have any other questions!

Here are five related questions to extend your understanding:
1. How would the solution change if the exponent on \( z^{-2} \) was different?
2. What happens if the entire expression is multiplied by another term like \( w^{-1} \)?
3. How would you handle the case where there is a negative exponent applied to the whole expression?
4. How does the distributive property of exponents apply in this problem?
5. What would happen if the variable \( w \) also had a negative exponent inside the parentheses?

**Tip:** When simplifying expressions with exponents, remember to apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\).

Learn how to simplify the algebraic expression (5w^2z^{-2})^2(z^3) step-by-step using exponent rules. This problem helps in understanding the power of a power rule and multiplying terms with the same base. Suitable for students in Grades 8-10.

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