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.