Skill: Evaluate powers with negative integer bases

A power with a negative integer base means repeated multiplication of the negative number. The parentheses show that the negative sign is part of the base.

For example, evaluate

$$(-2)^5$$

1. The base is $-2$, and the exponent is $5$. So multiply $-2$ five times:

$$(-2)^5=(-2)(-2)(-2)(-2)(-2)$$

2. Pair the factors:

$$(-2)(-2)=4$$

There are five negative factors, so one negative factor is left over. Therefore, the product is negative.

3. Multiply the absolute values:

$$2^5=32$$

Thus,

$$(-2)^5=-32$$

An even exponent gives a positive answer because the negative factors pair up. An odd exponent gives a negative answer. Also, any nonzero number raised to the zero power equals $1$:

$$(-2)^0=1$$

Be careful with parentheses:

$$(-2)^4=16$$

because the base is $-2$, but

$$-2^4=-16$$

because the exponent applies only to $2$, and the negative sign remains outside the power.