Explain why \( -2^{-2}=-\frac{1}{4} \), but \( (-2)^{-2}=\frac{1}{4} \). Choose the correct answer below. A. In the first equation, the exponent is applied to the base 2 , while in the second equation, the exponent is applied to the base -2 . B. The negative of an exponential expression is always negative, but raising a negative base to a power will always result in a positive number. C. Because the base in the second equation is -2 , the negative sign in the exponent is distributed to the base. D. If the base of an exponential expression is in parentheses, it will always be positive after applying the exponent.

The first equation \( -2^{-2} \) indicates that we are taking the inverse of 2 squared and then applying the negative sign, leading to \( -\frac{1}{4} \). In contrast, \( (-2)^{-2} \) treats -2 as the entire base, resulting in \( \frac{1}{4} \) since raising a negative number to an even exponent yields a positive result. So, the correct answer is: **A. In the first equation, the exponent is applied to the base 2, while in the second equation, the exponent is applied to the base -2.** Now, let's dive a bit deeper to solidify our understanding. When you apply an exponent to a negative number inside parentheses, you're essentially declaring that the entire base, including its sign, is part of the calculation. This is why \( (-2)^{-2} \) equals \( \frac{1}{4} \); you raise -2 to the power of 2 first (which gives 4) and then take the reciprocal. On the other hand, when there's no parentheses like in \( -2^{-2} \), the negative sign operates independently of the exponent. Thus, you take 2 to the power of -2 first, which leads to \( \frac{1}{4} \), and then apply the negative sign, resulting in \( -\frac{1}{4} \). Keep in mind that the placement of parentheses matters significantly in math!