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The expression you need to simplify and evaluate is:

\[
\frac{32x^2y^{-2}z^{-2}}{x^{-2}y^{-3}z^3}
\]
where \( x = 2 \), \( y = 4 \), and \( z = -2 \).

### Step 1: Simplify the expression algebraically

Use the properties of exponents to combine like terms. The general rules for simplifying are:

\[
\frac{a^m}{a^n} = a^{m-n}
\quad \text{and} \quad a^{-m} = \frac{1}{a^m}.
\]

So, for each variable:

- For \( x \): \( \frac{x^2}{x^{-2}} = x^{2 - (-2)} = x^{4} \).
- For \( y \): \( \frac{y^{-2}}{y^{-3}} = y^{-2 - (-3)} = y^{1} \).
- For \( z \): \( \frac{z^{-2}}{z^3} = z^{-2 - 3} = z^{-5} \).

The expression now becomes:

\[
32 \cdot x^4 \cdot y \cdot z^{-5}
\]

### Step 2: Substitute the given values

Now, substitute \( x = 2 \), \( y = 4 \), and \( z = -2 \) into the simplified expression:

\[
32 \cdot (2^4) \cdot 4 \cdot (-2)^{-5}
\]

### Step 3: Evaluate each term

- \( 2^4 = 16 \)
- \( (-2)^{-5} = \frac{1}{(-2)^5} = \frac{1}{-32} \)

So the expression becomes:

\[
32 \cdot 16 \cdot 4 \cdot \frac{1}{-32}
\]

### Step 4: Simplify the expression

- \( 32 \) and \( \frac{1}{-32} \) cancel out to give \( -1 \).
- Now you have:
\[
-1 \cdot 16 \cdot 4 = -64
\]

### Final Answer:
The value of the expression is \( -64 \).

Would you like a more detailed explanation, or do you have any further questions?

Here are five related questions you might want to explore:
1. How do negative exponents affect division of terms?
2. What are other exponent rules that apply in this problem?
3. How can fractional exponents be simplified in similar expressions?
4. Can the value of \( z^{-5} \) change if \( z \) is a fraction?
5. What happens when bases are negative in exponentiation?

**Tip:** Always simplify the exponents before substituting values to avoid errors in calculation!

Evaluate the expression \(\frac{32x^2y^{-2}z^{-2}}{x^{-2}y^{-3}z^3}\) when \(x = 2\), \(y = 4\), and \(z = -2\).

Learn how to evaluate and simplify expressions with negative exponents like \(\frac{32x^2y^{-2}z^{-2}}{x^{-2}y^{-3}z^3}\), using exponent rules. Substitute given values for x, y, and z to find the final result. Suitable for students in Grades 9-11.

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