https://file.aiquickdraw.com/mathbot/file/1725498584190kzm4ekwk.jpg

The expression you uploaded is:

\[
\left( \frac{5^{-2} \cdot 3^2}{3^3 \cdot 5^5} \right)^{-3}
\]

Let’s simplify the expression step by step.

### Step 1: Simplify the expression inside the parentheses.

We have:
\[
\frac{5^{-2} \cdot 3^2}{3^3 \cdot 5^5}
\]

- For the powers of 5:
\[
\frac{5^{-2}}{5^5} = 5^{-2-5} = 5^{-7}
\]

- For the powers of 3:
\[
\frac{3^2}{3^3} = 3^{2-3} = 3^{-1}
\]

Now, the expression becomes:
\[
5^{-7} \cdot 3^{-1}
\]

### Step 2: Apply the outer exponent of \(-3\).
We now have the following expression:
\[
\left( 5^{-7} \cdot 3^{-1} \right)^{-3}
\]

Using the power of a product rule:
\[
(5^{-7})^{-3} \cdot (3^{-1})^{-3} = 5^{21} \cdot 3^{3}
\]

### Step 3: Final expression.

The simplified form of the expression is:
\[
5^{21} \cdot 3^{3}
\]

This is the final answer.

Would you like to explore further details or have any questions?

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Here are 5 related questions:
1. How do we handle negative exponents in simplification?
2. What is the product of powers rule in exponents?
3. How can we simplify expressions with different bases?
4. What are the steps to simplify more complex fractions with exponents?
5. How do we apply exponentiation to a product?

**Tip:** Remember that when dividing powers with the same base, subtract the exponents!

Simplify the expression: \( \left( \frac{5^{-2} \cdot 3^2}{3^3 \cdot 5^5} \right)^{-3} \)

This guide walks through simplifying the expression \( \left( \frac{5^{-2} \cdot 3^2}{3^3 \cdot 5^5} \right)^{-3} \) using laws of exponents. Learn to apply the power of a product rule and handle negative exponents, suitable for Grades 9-11.

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