https://file.aiquickdraw.com/mathbot/file/1726125367856d8h6lscj.jpg

The expression visible in the image is:

\[
\frac{x^{-2}y^2z^3}{x^{-1}yz^{-1}} \times \left( \frac{(4^{-2})x^2yz}{x^3y^2z^{-1}} \right)^{-1}
\]

Let's simplify this step by step.

### Step 1: Simplify the first fraction
\[
\frac{x^{-2}y^2z^3}{x^{-1}yz^{-1}}
\]

- For the \(x\) terms: \(x^{-2} / x^{-1} = x^{-2 - (-1)} = x^{-1}\)
- For the \(y\) terms: \(y^2 / y = y^{2 - 1} = y\)
- For the \(z\) terms: \(z^3 / z^{-1} = z^{3 - (-1)} = z^4\)

Thus, the simplified first part is:
\[
x^{-1} y z^4
\]

### Step 2: Simplify the second fraction before inversion
\[
\frac{(4^{-2})x^2yz}{x^3y^2z^{-1}}
\]

- For the constants: \(4^{-2} = \frac{1}{16}\)
- For the \(x\) terms: \(x^2 / x^3 = x^{2 - 3} = x^{-1}\)
- For the \(y\) terms: \(y / y^2 = y^{1 - 2} = y^{-1}\)
- For the \(z\) terms: \(z / z^{-1} = z^{1 - (-1)} = z^2\)

Thus, this part simplifies to:
\[
\frac{1}{16} x^{-1} y^{-1} z^2
\]

### Step 3: Invert the second fraction
When we invert the second fraction, we flip it, so it becomes:
\[
16 x y z^{-2}
\]

### Step 4: Multiply both parts
Now, multiply the simplified expressions from Step 1 and Step 3:
\[
(x^{-1} y z^4) \times (16 x y z^{-2})
\]

- For the constants: \(1 \times 16 = 16\)
- For the \(x\) terms: \(x^{-1} \times x = x^{-1 + 1} = x^0 = 1\)
- For the \(y\) terms: \(y \times y = y^2\)
- For the \(z\) terms: \(z^4 \times z^{-2} = z^{4 - 2} = z^2\)

Thus, the final simplified expression is:
\[
16 y^2 z^2
\]

Let me know if you'd like any more details or clarifications!

Here are five related questions to explore:
1. What are the laws of exponents used in simplifying terms like \(x^{-1} \times x\)?
2. How do we handle negative exponents when simplifying fractions?
3. What is the process for inverting a fraction in multiplication?
4. How do constant factors (like \(4^{-2}\)) influence the overall expression?
5. How can this simplification process be applied to more complex algebraic expressions?

**Tip:** Always remember that multiplying and dividing terms with exponents follows simple addition and subtraction rules for the exponents.

Simplify the expression \(\frac{x^{-2}y^2z^3}{x^{-1}yz^{-1}} \times \left( \frac{(4^{-2})x^2yz}{x^3y^2z^{-1}} \right)^{-1}\)

This problem demonstrates how to simplify a complex expression using exponent rules, fraction simplification, and inversion. It includes step-by-step methods to combine terms with exponents, invert fractions, and apply mathematical rules. Suitable for high school students (Grades 9-12).

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