Compare Nested Exponents: (-(-3)³)² vs (2²)⁴

To solve this problem, we need to follow these steps:

• Step 1: Evaluate the first expression $(-(-3)^3)^2$
• Step 2: Evaluate the second expression $((2)^2)^4$
• Step 3: Compare the results from Steps 1 and 2

Now, let's proceed with these steps:

Step 1: Evaluate the expression $(-(-3)^3)^2$.
The inner expression is $(-3)^3$. Calculating this gives: $(-3)^3 = -27$ Next, we compute the expression $-(-3)^3$, which simplifies to: $-(-27) = 27$ Finally, we square this result: $(27)^2 = 729$ Thus, the value of the first expression is 729.

Step 2: Evaluate the expression $((2)^2)^4$.
First, calculate $(2)^2$: $(2)^2 = 4$ Next, raise this result to the fourth power: $(4)^4 = 256$ Thus, the value of the second expression is 256.

Step 3: Compare the two results from above:
We have $(-(-3)^3)^2 = 729$ and $((2)^2)^4 = 256$.

Since 729 is greater than 256, the expression $(-(-3)^3)^2$ is larger.

Thus, the correct answer is $\mathbf{>}$.