Solve (-(-2)²)² - 2³: Multiple Exponents and Negative Numbers

To solve this problem, we need to carefully apply the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Let's evaluate the expression step-by-step:

• Step 1: Begin with the expression inside the innermost parentheses: $(-2)^2$. This means we first square $-2$, which results in $4$ because $(-2) \times (-2) = 4$.
• Step 2: Now, consider the negative sign outside the squared term. So, evaluate $-(-2)^2$, which simplifies to $-4$.
• Step 3: Next, we need to square $-4$: $(-4)^2 = (-4) \times (-4) = 16$.
• Step 4: Now, calculate the second part of the expression, $2^3$, which equals $8$ because $2 \times 2 \times 2 = 8$.
• Step 5: Finally, subtract the result from Step 4 from the result of Step 3: $16 - 8 = 8$.

Thus, the solution to the problem is $8$.