https://file.aiquickdraw.com/mathbot/file/17292199271091krda38v.jpg

Certainly! The expression in the image is:

\[
\left( 4x^{-5} y^6 z^3 \right)^2
\]

This involves raising a product of terms to a power. To simplify this expression, we need to apply the power rule to each factor inside the parentheses. The power rule for exponents states:

\[
(a^m)^n = a^{m \cdot n}
\]

We will apply this rule to each of the terms inside the parentheses.

### Step-by-Step Solution:

1. **Apply the exponent to the constant**:
   \[
   4^2 = 16
   \]

2. **Apply the exponent to \( x^{-5} \)**:
   \[
   (x^{-5})^2 = x^{-10}
   \]

3. **Apply the exponent to \( y^6 \)**:
   \[
   (y^6)^2 = y^{12}
   \]

4. **Apply the exponent to \( z^3 \)**:
   \[
   (z^3)^2 = z^6
   \]

Now, multiplying everything together, the simplified expression becomes:

\[
16x^{-10}y^{12}z^6
\]

### Final Answer:
\[
16x^{-10}y^{12}z^6
\]

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

Here are five related questions for further practice:
1. How would you simplify \( \left( 3x^4y^{-2}z^3 \right)^3 \)?
2. What happens if the exponent inside the parentheses is a fraction instead of a whole number?
3. How can you simplify expressions with negative exponents, like \( x^{-5} \)?
4. Can you apply this method to expressions that have roots instead of exponents?
5. What would the result be if you multiplied this final expression by \( x^{10} \)?

**Tip**: Remember that negative exponents can be rewritten as fractions (e.g., \( x^{-n} = \frac{1}{x^n} \)).

Simplify the expression (4x^-5 y^6 z^3)^2.

Learn how to simplify the expression (4x^-5 y^6 z^3)^2 using exponentiation rules. This problem covers key algebraic concepts such as applying powers to variables and constants. Suitable for students in Grades 9-11.

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