Negative Bases

A negative number can be a base too, written in parentheses: ${\left(-5\right)}^{2}$ means $\left(-5\right) \cdot \left(-5\right)$. Two negatives multiply to a positive, so ${\left(-5\right)}^{2}$ is 25.

Add one more factor: ${\left(-5\right)}^{3}$ is $\left(-5\right) \cdot \left(-5\right) \cdot \left(-5\right)$. Multiply one pair at a time. The first pair gives 25, so the next line is $25 \cdot \left(-5\right)$, which is -125. The third negative factor had no partner, so it turned the result negative again.

That gives a rule you can read straight off the exponent. Negative factors pair up, and each pair makes a positive. With an even exponent every factor has a partner, so the result is positive. With an odd exponent one negative is left over, so the result is negative.