Let's simplify the expression $\frac{-4^4\sqrt{32}}{2^4\sqrt{2}}$.

Step-by-step solution:

1. Simplify the powers:

   $-4^4 = -(4^4) = -256$

   $2^4 = 16$

   Now, the expression becomes: $\frac{-256\sqrt{32}}{16\sqrt{2}}$

2. Simplify the square roots:

   • $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$
   • $\sqrt{2} = \sqrt{2}$

   Substitute these values: $\frac{-256 \times 4\sqrt{2}}{16 \times \sqrt{2}} = \frac{-1024 \sqrt{2}}{16 \sqrt{2}}$

3. Cancel out common terms:
   The $\sqrt{2}$ in the numerator and denominator cancels out, leaving: $\frac{-1024}{16}$

4. Perform the division: $\frac{-1024}{16} = -64$

Final Answer:

$\boxed{-64}$

Would you like more details or explanations about any step?

Here are 5 related questions:

1. What is the general rule for simplifying square roots of products?
2. How do you simplify fractional exponents?
3. How can you simplify expressions involving negative powers?
4. What is the meaning of rationalizing a denominator in square root problems?
5. How can you use exponent rules to simplify complex expressions?

Tip: Always simplify roots and powers before performing multiplications or divisions for easier calculations.