Can you simplify an expression where an exponent has an exponent? For example, how would you simplify ?

### Watch This

James Sousa: Properties of Exponents

### Guidance

In the expression
, the
is called the
**
base
**
and the
is called the
**
exponent
**
.
**
Exponents
**
are often referred to as
**
powers
**
. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

- .

There are many rules that have to do with exponents (often called the
**
Laws of Exponents
**
) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn a rule that has to do with raising a power to another power.

**
RULE: To raise a power to a new power, multiply the exponents.
**

#### Example A

Evaluate .

**
Solution:
**
.

#### Example B

Simplify .

**
Solution:
**
.

#### Example C

Evaluate .

**
Solution:
**
.

#### Example D

Simplify .

**
Solution:
**
.

#### Concept Problem Revisited

. Notice that the power rule applies even when a number has been raised to more than one power. The overall exponent is 24 which is .

### Guided Practice

You know you can rewrite as and then calculate in order to find that . This concept can also be reversed. To write 32 as a power of 2, . There are 5 twos; therefore, . Use this idea to complete the following problems.

1. Write 81 as a power of 3.

2. Write as a power of 3.

3. Write as a power of 2.

**
Answers:
**

1.

- There are 4 threes. Therefore

2.

- There are 2 threes. Therefore .
- Apply the law of exponents for power to a power-multiply the exponents.
- Therefore

3.

- There are 2 twos. Therefore
- Apply the law of exponents for power to a power-multiply the exponents.
- Apply the law of exponents for power to a power-multiply the exponents.
- Therefore

### Explore More

Simplify each of the following expressions.

- True or false:
- True or false:
- Write 64 as a power of 4.
- Write as a power of 2.
- Write as a power of 3.
- Write as a power of 3.
- Write as a power of 5.