# 8.3: Exponents and Roots

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Learning Objectives

• understand and be able to read exponential notation
• understand the concept of root and be able to read root notation
• be able to use a calculator having the $$y^x$$ key to determine a root

## Exponential Notation

Definition: Exponential Notation

We have noted that multiplication is a description of repeated addition. Exponen­tial notation is a description of repeated multiplication.

Suppose we have the repeated multiplication

$$8 \cdot 8 \cdot 8 \cdot 8 \cdot 8$$

Definition: Exponent

The factor 8 is repeated 5 times. 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.

Definition: The Function of an Exponent

An exponent records the number of identical factors that are repeated in a multiplication.

Sample Set A

Write the following multiplication using exponents.

$$3 \cdot 3$$. Since the factor 3 appears 2 times, we record this as

Solution

$$3^2$$

Sample Set A

$$62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62$$. Since the factor 62 appears 9 times, we record this as

Solution

$$62^9$$

Expand (write without exponents) each number.

Sample Set A

$$12^4$$. The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,

Solution

$$12^4 = 12 \cdot 12 \cdot 12 \cdot 12$$

Sample Set A

$$706^3$$. The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,

Solution

$$706^3 = 706 \cdot 706 \cdot 706$$

Practice Set A

Write the following using exponents.

$$37 \cdot 37$$

$$37^2$$

Practice Set A

$$16 \cdot 16 \cdot 16 \cdot 16 \cdot 16$$

$$16^5$$

Practice Set A

$$9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9$$

$$9^{10}$$

Write each number without exponents.

Practice Set A

$$85^3$$

$$85 \cdot 85 \cdot 85$$

Practice Set A

$$4^7$$

$$4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4$$

Practice Set A

$$1,739^2$$

$$1,739 \cdot 1,739$$

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 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.

Cubed
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 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.

### Roots

In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication.

Square Root
We know that $$49 = 7^2$$, that is, $$49 = 7 \cdot 7$$. Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49.

Cube Root
We know that $$8 = 2^3$$, that is, $$8 = 2 \cdot 2 \cdot 2$$. Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.

We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.

There is a symbol used to indicate roots of a number. It is called the radical sign $$\sqrt[n]{\ \ \ \ }$$

The Radical Sign $$\sqrt[n]{\ \ \ \ }$$

The symbol $$\sqrt[n]{\ \ \ \ }$$ is called a radical sign and indicates the nth root of a number.

We discuss particular roots using the radical sign as follows:

Square Root
$$\sqrt{\text{number}}$$ indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol $$\sqrt{\ \ \ \ }$$ is understood to be the square root radical sign.
$$\sqrt{49} = 7$$ since $$7 \cdot 7 = 7^2 = 49$$

Cube Root
$$\sqrt{\text{number}}$$ indicates the cube root of the number under the radical sign.
$$\sqrt{8} = 2$$ since $$2 \cdot 2 \cdot 2 = 2^3 = 8$$

Fourth Root
$$\sqrt{\text{number}}$$ indicates the fourth root of the number under the radical sign.
$$\sqrt{81} = 3$$ since $$3 \cdot 3 \cdot 3 \cdot 3 = 3^4 = 81$$
In an expression such as $$\sqrt{32}$$

$$\sqrt{\ \ \ \ }$$ is called the radical sign.

Index
5 is called the index. (The index describes the indicated root.)

$$\sqrt{32}$$ is called a radical (or radical expression).

Sample Set B

Find each root.

$$\sqrt{25}$$ To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,

Solution

$$\sqrt{25} = 5$$

Check: $$5 \cdot 5 = 5^2 = 25$$

Sample Set B

$$\sqrt{32}$$ To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

Solution

$$\sqrt{32} = 2$$

Check: $$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 = 32$$

Practice Set B

Find the following roots using only a knowledge of multiplication.

$$\sqrt{64}$$

8

Practice Set B

$$\sqrt{100}$$

10

Practice Set B

$$\sqrt{64}$$

4

Practice Set B

$$\sqrt{64}$$

2

## Calculators

Calculators with the $$\sqrt{x}$$, $$y^x$$, and $$1/x$$ keys can be used to find or approximate roots.

Sample Set C

Use the calculator to find $$\sqrt{121}$$

Solution

 Display Reads Type 121 121 Press $$\sqrt{x}$$ 11

Sample Set C

Find $$\sqrt{2187}$$.

Solution

 Display Reads Type 2187 2187 Press $$y^x$$ 2187 Type 7 7 Press $$1/x$$ .14285714 Press = 3

$$\sqrt{2187} = 3$$ (which means that $$3^7 = 2187$$)

Practice Set C

Use a calculator to find the following roots.

$$\sqrt{729}$$

9

Practice Set C

$$\sqrt{8503056}$$

54

Practice Set C

$$\sqrt{53361}$$

231

Practice Set C

$$\sqrt{16777216}$$

4

## Exercises

For the following problems, write the expressions using expo­nential notation.

Exercise $$\PageIndex{1}$$

$$4 \cdot 4$$

$$4^2$$

Exercise $$\PageIndex{2}$$

$$12 \cdot 12$$

Exercise $$\PageIndex{3}$$

$$9 \cdot 9 \cdot 9 \cdot 9$$

$$9^4$$

Exercise $$\PageIndex{4}$$

$$10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$$

Exercise $$\PageIndex{5}$$

$$826 \cdot 826 \cdot 826$$

$$826^3$$

Exercise $$\PageIndex{6}$$

$$3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021$$

Exercise $$\PageIndex{7}$$

$$\begin{matrix} \underbrace{6 \cdot 6 \cdots\cdots 6} \\ {\text{85 factors of 6}} \end{matrix}$$

$$6^{85}$$

Exercise $$\PageIndex{8}$$

$$\begin{matrix} \underbrace{2 \cdot 2 \cdots\cdots 2} \\ {\text{112 factors of 2}} \end{matrix}$$

Exercise $$\PageIndex{9}$$

$$\begin{matrix} \underbrace{1 \cdot 1 \cdots\cdots 1} \\ {\text{3,008 factors of 1}} \end{matrix}$$

$$1^{3008}$$

For the following problems, expand the terms. (Do not find the actual value.)

Exercise $$\PageIndex{10}$$

$$5^3$$

Exercise $$\PageIndex{11}$$

$$7^4$$

$$7 \cdot 7 \cdot 7 \cdot 7$$

Exercise $$\PageIndex{12}$$

$$15^2$$

Exercise $$\PageIndex{13}$$

$$117^5$$

$$117 \cdot 117 \cdot 117 \cdot 117 \cdot 117$$

Exercise $$\PageIndex{14}$$

$$61^6$$

Exercise $$\PageIndex{15}$$

$$30^2$$

$$30 \cdot 30$$

For the following problems, determine the value of each of the powers. Use a calculator to check each result.

Exercise $$\PageIndex{16}$$

$$3^2$$

Exercise $$\PageIndex{17}$$

$$4^2$$

$$4 \cdot 4 = 16$$

Exercise $$\PageIndex{18}$$

$$1^2$$

Exercise $$\PageIndex{19}$$

$$10^2$$

$$10 \cdot 10 = 100$$

Exercise $$\PageIndex{20}$$

$$11^2$$

Exercise $$\PageIndex{21}$$

$$12^2$$

$$12 \cdot 12 = 144$$

Exercise $$\PageIndex{22}$$

$$13^2$$

Exercise $$\PageIndex{23}$$

$$15^2$$

$$15 \cdot 15 = 225$$

Exercise $$\PageIndex{24}$$

$$1^4$$

Exercise $$\PageIndex{25}$$

$$3^4$$

$$3 \cdot 3 \cdot 3 \cdot 3 = 81$$

Exercise $$\PageIndex{26}$$

$$7^3$$

Exercise $$\PageIndex{27}$$

$$10^3$$

$$10 \cdot 10 \cdot 10 = 1000$$

Exercise $$\PageIndex{28}$$

$$100^2$$

Exercise $$\PageIndex{29}$$

$$8^3$$

$$8 \cdot 8 \cdot 8 = 512$$

Exercise $$\PageIndex{30}$$

$$5^5$$

Exercise $$\PageIndex{31}$$

$$9^3$$

$$9 \cdot 9 \cdot 9 = 729$$

Exercise $$\PageIndex{32}$$

$$6^2$$

Exercise $$\PageIndex{33}$$

$$7^1$$

$$7^1 = 7$$

Exercise $$\PageIndex{34}$$

$$1^{28}$$

Exercise $$\PageIndex{35}$$

$$2^7$$

$$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 128$$

Exercise $$\PageIndex{36}$$

$$0^5$$

Exercise $$\PageIndex{37}$$

$$8^4$$

$$8 \cdot 8 \cdot 8 \cdot 8 = 4,096$$

Exercise $$\PageIndex{38}$$

$$5^8$$

Exercise $$\PageIndex{39}$$

$$6^9$$

$$6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 10,077,696$$

Exercise $$\PageIndex{40}$$

$$25^3$$

Exercise $$\PageIndex{41}$$

$$42^2$$

$$42 \cdot 42 = 1,764$$

Exercise $$\PageIndex{42}$$

$$31^3$$

Exercise $$\PageIndex{43}$$

$$15^5$$

$$15 \cdot 15 \cdot 15 \cdot 15 \cdot 15 = 759,375$$

Exercise $$\PageIndex{44}$$

$$2^{20}$$

Exercise $$\PageIndex{45}$$

$$816^2$$

$$816 \cdot 816 = 665,856$$

For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.

Exercise $$\PageIndex{46}$$

$$\sqrt{9}$$

Exercise $$\PageIndex{47}$$

$$\sqrt{16}$$

4

Exercise $$\PageIndex{48}$$

$$\sqrt{36}$$

Exercise $$\PageIndex{49}$$

$$\sqrt{64}$$

8

Exercise $$\PageIndex{50}$$

$$\sqrt{121}$$

Exercise $$\PageIndex{51}$$

$$\sqrt{144}$$

12

Exercise $$\PageIndex{52}$$

$$\sqrt{169}$$

Exercise $$\PageIndex{53}$$

$$\sqrt{225}$$

15

Exercise $$\PageIndex{54}$$

$$\sqrt{27}$$

Exercise $$\PageIndex{55}$$

$$\sqrt{32}$$

2

Exercise $$\PageIndex{56}$$

$$\sqrt{1}$$

Exercise $$\PageIndex{57}$$

$$\sqrt{400}$$

20

Exercise $$\PageIndex{58}$$

$$\sqrt{900}$$

Exercise $$\PageIndex{59}$$

$$\sqrt{10,000}$$

100

Exercise $$\PageIndex{60}$$

$$\sqrt{324}$$

Exercise $$\PageIndex{61}$$

$$\sqrt{3,600}$$

60

For the following problems, use a calculator with the keys $$\sqrt{x}$$, $$y^x$$, and $$1/x$$ to find each of the values.

Exercise $$\PageIndex{62}$$

$$\sqrt{676}$$

Exercise $$\PageIndex{63}$$

$$\sqrt{1,156}$$

34

Exercise $$\PageIndex{64}$$

$$\sqrt{46,225}$$

Exercise $$\PageIndex{65}$$

$$\sqrt{17,288,964}$$

4,158

Exercise $$\PageIndex{66}$$

$$\sqrt{3,375}$$

Exercise $$\PageIndex{67}$$

$$\sqrt{331,776}$$

24

Exercise $$\PageIndex{68}$$

$$\sqrt{5,764,801}$$

Exercise $$\PageIndex{69}$$

$$\sqrt{16,777,216}$$

4

Exercise $$\PageIndex{70}$$

$$\sqrt{16,777,216}$$

Exercise $$\PageIndex{71}$$

$$\sqrt{9,765,625}$$

5

Exercise $$\PageIndex{72}$$

$$\sqrt{160,000}$$

Exercise $$\PageIndex{73}$$

$$\sqrt{531,441}$$

81

#### Exercises for Review

Exercise $$\PageIndex{74}$$

Use the numbers 3, 8, and 9 to illustrate the associative property of addition.

Exercise $$\PageIndex{75}$$

In the multiplication $$8 \cdot 4 = 32$$, specify the name given to the num­bers 8 and 4.

81

Exercise $$\PageIndex{76}$$

Does the quotient $$15 \div 0$$ exist? If so, what is it?

Exercise $$\PageIndex{77}$$

Does the quotient $$0 \div 15$$ exist? If so, what is it?

Exercise $$\PageIndex{78}$$