Example 1: 2³

First, let's solve: 2³

The exponent tells us how many times to multiply the base by itself. In this example, the base is 2 and the exponent is 3.

2³ = 2 × 2 × 2

2 × 2 = 4

4 × 2 = 8

Answer: 2³ = 8

Example 2: (2/3)³

Now let's solve: (2/3)³

When a fraction is raised to a power, we raise both the numerator and denominator to that power.

(2/3)³ = 2³ / 3³

2³ = 8

3³ = 3 × 3 × 3 = 27

Answer: 8/27

Remember:

Base = the number being multiplied Exponent = how many times to multiply it

Example 3: 3⁰

Now let's look at: 3⁰

There is a special rule:

Any nonzero number raised to the power of zero equals 1.

3⁰ = 1

7⁰ = 1

2⁰ = 1

Answer: 3⁰ = 1

Example 4: 2^-3

Now let's solve: 2^-3

A negative exponent means we take the reciprocal of the base.

2 = 2/1

Reciprocal = 1/2

(1/2)³

1³ / 2³

1 / 8

Answer: 1/8

Example 5: (1/2)^-3

Now let's solve: (1/2)^-3

Since the exponent is negative, we take the reciprocal.

2/1

2³

2 × 2 × 2 = 8

Answer: 8

Example 6: (3/4)^-2

Let's solve: (3/4)^-2

First, take the reciprocal:

4/3

Then square both the numerator and denominator:

4² / 3²

16 / 9

Answer: 16/9

As a mixed number:

1 7/9

Challenge Problem: 4^3/2

Solve: 4^3/2

This exponent is called a fractional exponent.

3/2 = 1 + 1/2

4^{1 + 1/2}

4¹ × 4^1/2

4 × √4

4 × 2

8

Answer: 8