1.4: Exponents

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Dec 21, 2021
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- 1.3.4.1: Exercises
- 1.4.1: Rules of Exponents

Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
Rio Hondo College

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1.4.1 Learning Objectives

Read, interpret, and write exponential notation

Exponential Notation

Definition: Exponential Notation

Just as multiplication is a description of repeated addition, exponential notation is a description of repeated multiplication.

Suppose we have the repeated multiplication $8 \cdot 8 \cdot 8 \cdot 8 \cdot 8$ .

The factor 8 is repeated 5 times.

Definition: Exponent

Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, $8^5$ , in this case. The superscript is called an exponent. An exponent records the number of identical factors that are repeated in a multiplication.

If $x$ is any real number and $n$ is a natural number, then
$x^{n}=\underbrace{x \cdot x \cdot x \cdot \ldots \cdot x}_{n \text { factors of } x}$
An exponent records the number of identical factors in a multiplication.

Example 1

Write the following multiplications using exponents.

$3 \cdot 3$
$62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62$

Solution

Since the factor 3 appears 2 times, we record this as $3^2$ .
Since the factor 62 appears 9 times, we record this as $62^9$ .

Example 3

Expand (write without exponents) each number.

$12^4$
$706^3$
$15^1$

Solution

The exponent 4 is recording 4 factors of 12 in a multiplication. Thus, $12^4 = 12 \cdot 12 \cdot 12 \cdot 12$ .
The exponent 3 is recording 3 factors of 706 in a multiplication. Thus, $706^3 = 706 \cdot 706 \cdot 706$ .
The exponent 1 is recording 1 factor of 15 in a multiplication. Thus, $15^1 = 15$ .

Try It Now 1

Write the following using exponents.

$37 \cdot 37$

Answer: $37^2$

Reading Exponential Notation

Base Exponent Power

In $x^n$ ,

$x$ is the base
$n$ is the exponent
The number represented by $x^n$ is called a power

The term $x^n$ is read as " $x$ to the $n$ th."

In a number such as $8^5$ .

Base
8 is called the base.

Exponent, Power
5 is called the exponent, or power. $8^5$ is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."

Squared
When a 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.

Cubed
When a 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 number is raised to the power of 4 or higher, we simply say that that 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.

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Text Color

Text Size

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Base Exponent Power

Exponential Notation

Reading Exponential Notation

Base Exponent Power

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