Compare (-4)^2)^2 vs -(4^2)^2: Exponent Order Challenge

To solve this problem, we must calculate both expressions step-by-step:

First, consider the expression $(( -4)^2)^2$:

• Evaluate $(-4)^2$:
  $(-4)^2 = (-4) \times (-4) = 16$.
• Now, evaluate $16^2$:
  $16^2 = 16 \times 16 = 256$.

The value of $(( -4)^2)^2$ is $256$.

Next, consider the expression $(-(4)^2)^2$:

• Evaluate $(4)^2$:
  $4^2 = 4 \times 4 = 16$.
• Now, consider the negative sign: $-(4)^2 = -(16) = -16$.
• Evaluate $(-16)^2$:
  $(-16)^2 = (-16) \times (-16) = 256$.

The value of $(-(4)^2)^2$ is $256$.

Therefore, the two expressions are equal.

Conclusion: The correct choice is $(=)$.