The expression in the image is:

$\frac{(16x^{-4})^2}{(4x^{-2})^3}$

Let's simplify this step-by-step.

Step 1: Apply the power rule $(a^m)^n = a^{m \cdot n}$

For the numerator: $(16x^{-4})^2 = 16^2 \cdot x^{-4 \cdot 2} = 256x^{-8}$

For the denominator: $(4x^{-2})^3 = 4^3 \cdot x^{-2 \cdot 3} = 64x^{-6}$

Now the expression becomes: $\frac{256x^{-8}}{64x^{-6}}$

Step 2: Simplify the constants

$\frac{256}{64} = 4$

So the expression is now: $4 \cdot \frac{x^{-8}}{x^{-6}}$

Step 3: Apply the quotient rule for exponents $\frac{a^m}{a^n} = a^{m-n}$

$\frac{x^{-8}}{x^{-6}} = x^{-8 - (-6)} = x^{-8 + 6} = x^{-2}$

Thus, the expression simplifies to: $4x^{-2}$

Final Answer:

$4x^{-2}$

If you'd like, I can explain any step in further detail.

Would you like more details, or do you have any questions?

Here are 5 questions to consider:

1. How do power rules apply to negative exponents?
2. Can you explain the difference between multiplying and dividing powers?
3. How does raising a term to a power affect both the base and exponent?
4. Why does $x^{-2}$ represent a reciprocal, and what is its equivalent?
5. What happens when both the numerator and denominator have negative exponents?

Tip: When simplifying expressions with exponents, always apply exponent rules methodically to avoid mistakes.