Skill: Interpret powers with whole-number exponents

A power shows repeated multiplication.

In $a^n$:

• $a$ is the base.
• $n$ is the exponent.
• The exponent tells how many times to use the base as a factor.

For example, $3^4$ means

$$3^4=3\cdot3\cdot3\cdot3=81.$$

When the base is negative, parentheses matter. Let’s evaluate and compare:

$$(-2)^3,\qquad -2^3,\qquad (-2)^4,\qquad 5^0$$

Step 1: Evaluate $(-2)^3$

The parentheses show that the base is $-2$:

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

First, $(-2)(-2)=4$. Then $4(-2)=-8$, so

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

An odd number of negative factors gives a negative product.

Step 2: Evaluate $-2^3$

Without parentheses, the exponent applies only to $2$:

$$-2^3=-(2^3)=-(2\cdot2\cdot2)=-8.$$

Here, the negative sign is outside the power.

Step 3: Evaluate $(-2)^4$

The base is again $-2$:

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

An even number of negative factors gives a positive product.

Step 4: Evaluate $5^0$

Any nonzero number raised to the zero power equals $1$:

$$5^0=1.$$

So the answers are

$$(-2)^3=-8,\qquad -2^3=-8,\qquad (-2)^4=16,\qquad 5^0=1.$$

Remember: $(-a)^n$ raises the entire negative number to a power, while $-a^n$ means the negative of $a^n$.