Skill: Distinguish negative powers from powers with negative bases

A negative exponent tells you to take a reciprocal. It does not make the base negative:

$$a^{-n}=\frac{1}{a^n}$$

Parentheses show whether the negative sign is part of the base.

Example: Evaluate each expression.

1. $2^{-3}$

The base is positive $2$, and the exponent is negative. Take the reciprocal:

$$2^{-3}=\frac{1}{2^3}=\frac18$$

So, $2^{-3}$ is positive.

2. $(-2)^3$

The base is negative because the negative sign is inside the parentheses. The exponent is odd, so the answer is negative:

$$(-2)^3=(-2)(-2)(-2)=-8$$

3. $(-2)^4$

Again, the base is negative, but the exponent is even. A negative number multiplied an even number of times gives a positive result:

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

4. $-2^2$

There are no parentheses around $-2$, so the exponent applies only to $2$. The subtraction sign is applied afterward:

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

Remember:

• A negative exponent means reciprocal: $2^{-3}=\frac18$.
• A negative base must be shown with parentheses: $(-2)^3$.
• For a negative base, an odd power is negative and an even power is positive.
• $(-2)^2$ and $-2^2$ are different: $4$ versus $-4$.