Skill: Evaluate powers with integer exponents

A power tells you how many times to use the base as a factor.

• For a positive exponent, multiply the base repeatedly:
  $\displaystyle a^n=\underbrace{a\cdot a\cdot \ldots \cdot a}_{n\text{ factors}}$
• For a nonzero base raised to the zero power, $a^0=1$.
• For a negative exponent, take the reciprocal, then use the positive exponent:
  $\displaystyle a^{-n}=\frac{1}{a^n}$

Pay attention to parentheses. In $\left(-0.5\right)^3$, the negative number is the base. The sign of the answer depends on the base and whether the exponent is even or odd.

Example

Evaluate:

$$\left(-0.5\right)^{-3}$$

Step 1: Use the negative-exponent rule.

Take the reciprocal and change the exponent to positive:

$$\left(-0.5\right)^{-3}=\frac{1}{\left(-0.5\right)^3}$$

Step 2: Evaluate the positive power.

Since the exponent is $3$, multiply the base three times:

$$\left(-0.5\right)^3=(-0.5)(-0.5)(-0.5)=-0.125$$

There are three negative factors, so the product is negative.

Step 3: Take the reciprocal.

$$\frac{1}{-0.125}=-8$$

Therefore,

$$\boxed{\left(-0.5\right)^{-3}=-8}$$

The negative exponent caused the reciprocal; it did not make the answer negative. The answer is negative because the base is negative and the positive exponent $3$ is odd.