Exponential notation is a short way of writing the same number multiplied by itself many times.

Exponential notation uses a superscript for the number of times the number is repeated. The superscript is placed on the number to be multiplied (the factor), and is written like
*n* is an integer and *a* can be any real number. *a* is called the base and *n* is called the exponent or power.

The *n*th power of *a* is defined as:

*n* times)

with *a* multiplied by itself *n* times.

The resulting value is called the argument.

For example, instead of

5 is the base, and 6 is the exponent or power.

The result, 15625, is the argument.

5^{6} is read as “five to the sixth power,” or more simply as “five to the sixth,” or “the sixth power of five.”

Likewise

When a whole number is raised to the second power, it is said to be squared. The number 5^{2} can be read as

- 5 to the second power, or
- 5 to the second, or
- 5 squared.

When a whole number is raised to the third power, it is said to be cubed. The number 5^{3} can be read as

- 5 to the third power, or
- 5 to the third, or
- 5 cubed.

When a whole number is raised to the power of 4 or higher, we simply say that the number is raised to that particular power. The number 5^{8} can be read as

- 5 to the eighth power, or just
- 5 to the eighth.

We can also define what it means if we have a negative index, -*n*. Then,

*n* times)

If *n* is an even integer, then
*a*. For example, although -2 is negative,

**Examples, Exponential Notation**

Write the following multiplication using exponents:

#### Example 1

3 · 3

Since the factor 3 appears 2 times, we write this as

3^{2}

#### Example 2

62 · 62 · 62 · 62 · 62 · 62 · 62 · 62 · 62

Since the factor 62 appears nine times, we write this as:

62^{9}

Expand each number (write without exponents):

#### Example 3

12^{4}. The exponent 4 indicates that the base (12) is repeated 4 times, thus:

12^{4} = 12 · 12 · 12 · 12

#### Example 4

706^{3}. The exponent 3 indicates that the base (706) is repeated 3 times in a multiplication.

706^{3} = 706 · 706 · 706

**Exercises, Exponential Notation**

Write each of the following using exponents:

#### Exercise 1

37 · 37

##### Solution

37^{2}

#### Exercise 2

16 · 16 · 16 · 16 · 16

##### Solution

16^{5}

#### Exercise 3

9 · 9 · 9 · 9 · 9 · 9 · 9 · 9 · 9 · 9

##### Solution

9^{10}

Write each of the following numbers without exponents:

#### Exercise 4

85^{3}

##### Solution

85 · 85 · 85

#### Exercise 5

4^{7}

##### Solution

4 · 4 · 4 · 4 · 4 · 4 · 4

#### Exercise 6

1739^{2}

##### Solution

1739 · 1739