Powers of Negative Numbers

A negative base follows the same factor counting, plus the sign rules you already know. (-2)^2 is (-2) * (-2), two negatives, so the result is positive 4. (-2)^3 multiplies one more negative in, flipping the sign: -8.

Each extra factor of a negative number flips the sign once, so the pattern is decided by the exponent alone: an even exponent gives a positive result, an odd exponent keeps the result negative. (-1)^100 is 1; (-1)^101 is -1.

The size of the answer ignores the sign entirely: (-2)^5 has the same size as 2^5, namely 32. Work out the size by counting factors, then let the parity of the exponent choose the sign.