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Apply Laws of Exponents to write an equivalent expression for the given expression.

Let's analyze the given expression step by step using the laws of exponents.

The given equation is:

\[
\left( \left( -\frac{1}{3} w^{-4} z^{-1} \right)^{-2} \right) = \left( \left( -\frac{1}{3} \right)^{-2} \cdot \boxed{\phantom{w}} \cdot z^8 \right)
\]

### Step 1: Apply the exponent to each term inside the parentheses.
According to the power of a product rule, \((a \cdot b)^n = a^n \cdot b^n\), we will raise each part of the product \(\left( -\frac{1}{3} w^{-4} z^{-1} \right)\) to the power of \(-2\).

1. **First term** \(\left( -\frac{1}{3} \right)^{-2}\):
   \[
   \left( -\frac{1}{3} \right)^{-2} = \left( \frac{-1}{3} \right)^{-2} = \left( \frac{3}{-1} \right)^2 = 3^2 = 9
   \]

2. **Second term** \(w^{-4}\):
   Using the power rule \((a^m)^n = a^{m \cdot n}\), we have:
   \[
   (w^{-4})^{-2} = w^{(-4) \cdot (-2)} = w^8
   \]

3. **Third term** \(z^{-1}\):
   Similarly, we apply the power rule here:
   \[
   (z^{-1})^{-2} = z^{(-1) \cdot (-2)} = z^2
   \]

### Step 2: Combine the results.
Now, we multiply these results together:

\[
9 \cdot w^8 \cdot z^2
\]

### Step 3: Compare with the right-hand side.
On the right-hand side, we already see \( \left( -\frac{1}{3} \right)^{-2} = 9 \) and \(z^8\), so the missing boxed terms correspond to \(w^8\) and \(z^2\).

Thus, the completed expression becomes:

\[
9 w^8 z^2 = 9 w^8 z^8
\]

So the boxed terms should be \(w^8\) and \(z^2\).

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

Here are 5 related questions to expand on this topic:
1. What is the power of a product rule in exponents?
2. How do negative exponents affect the base?
3. What happens when you apply an exponent to a negative number?
4. Can you explain the steps to simplify complex exponentials?
5. What is the significance of combining like terms after exponentiation?

**Tip**: When dealing with negative exponents, remember that \(a^{-n} = \frac{1}{a^n}\).

Apply Laws of Exponents to write an equivalent expression for the given expression: \(\left( -\frac{1}{3} w^{-4} z^{-1} \right)^{-2} = \left( -\frac{1}{3} \right)^{-2} w^{8} z^{8}\)

Learn how to simplify expressions using the laws of exponents, focusing on applying the power of a product and power of a power rules. This step-by-step solution shows how to simplify \(\left( -\frac{1}{3} w^{-4} z^{-1} \right)^{-2}\) to an equivalent expression, suitable for Grades 8-10.

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