Express 64 as Different Mathematical Representations

$64=$

To solve this problem and express 64 as a power involving a negative number, we will follow these steps:

• Step 1: Recognize that we need to represent 64 using a base number squared. Since 64 is a perfect square, let's consider negative integers whose square equals 64.
• Step 2: The principal positive square root of 64 is 8. However, we are tasked with finding a negative number such that its square is 64.
• Step 3: If we have a negative integer, $(-8)$, and square it, we have: $(-8)^2 = (-8) \times (-8) = 64$.
• Step 4: Compare this with the expression $-(8)^2$, which results in $-64$ because the square applies only to 8, and the negative sign flips the result.

Therefore, the correct expression representing 64 with a negative base is $(-8)^2$, and among the answer choices provided, choice 1 is the correct one.