Let's first simplify the problem shown in the image step-by-step and confirm the answer:

The given expression is:

$\left( \frac{xy^4}{x^{-3}y} \right)^{-2}$

Step 1: Simplify inside the parentheses

Combine the $x$-terms and $y$-terms using the laws of exponents:

• For $x$: $\frac{x}{x^{-3}} = x^{1 - (-3)} = x^4$
• For $y$: $\frac{y^4}{y} = y^{4 - 1} = y^3$

So, the expression simplifies to:

$\left( x^4 y^3 \right)^{-2}$

Step 2: Apply the exponent $-2$

Distribute the $-2$ to each term:

• For $x^4$: $(x^4)^{-2} = x^{4 \cdot (-2)} = x^{-8}$
• For $y^3$: $(y^3)^{-2} = y^{3 \cdot (-2)} = y^{-6}$

So, the expression becomes:

$x^{-8} y^{-6}$

Step 3: Write with positive exponents

Using the property $a^{-n} = \frac{1}{a^n}$, rewrite the terms: $x^{-8} y^{-6} = \frac{1}{x^8 y^6}$

Thus, the simplified result is:

$\boxed{\frac{1}{x^8 y^6}}$

Answer:

The correct choice is C.

Four More Problems:

1. Simplify:
   $\left( \frac{x^2 y^{-3}}{x^{-1} y^2} \right)^3$

2. Simplify:
   $\left( \frac{a^5 b^{-2}}{a^{-3} b^3} \right)^{-1}$

3. Expand and simplify:
   $(2x^2 y^{-1})^3 \cdot (x^{-3} y^2)^2$

4. Simplify:
   $\left( \frac{m^2 n^{-4}}{m^{-3} n^5} \right)^{-2}$

Tip:

Always simplify the terms inside parentheses first before applying the outer exponents.