# 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.

\(37 \cdot 37\)

**Answer**-
\(37^2\)

Practice Set A

\(16 \cdot 16 \cdot 16 \cdot 16 \cdot 16\)

**Answer**-
\(16^5\)

Practice Set A

\(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\)

**Answer**-
\(9^{10}\)

Write each number without exponents.

Practice Set A

\(85^3\)

**Answer**-
\(85 \cdot 85 \cdot 85\)

Practice Set A

\(4^7\)

**Answer**-
\(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\)

Practice Set A

\(1,739^2\)

**Answer**-
\(1,739 \cdot 1,739\)

## Reading Exponential Notation

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.

## Reading Root Notation

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**

\(\sqrt[2]{\text{number}}\) 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**

\(\sqrt[3]{\text{number}}\) indicates the **cube root** of the number under the radical sign.

\(\sqrt[3]{8} = 2\) since \(2 \cdot 2 \cdot 2 = 2^3 = 8\)

**Fourth Root**

\(\sqrt[4]{\text{number}}\) indicates the **fourth root** of the number under the radical sign.

\(\sqrt[4]{81} = 3\) since \(3 \cdot 3 \cdot 3 \cdot 3 = 3^4 = 81\)

In an expression such as \(\sqrt[5]{32}\)

**Radical Sign**

\(\sqrt{\ \ \ \ }\) is called the **radical sign**.

**Index**

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

**Radicand**

32 is called the **radicand**.

**Radical**

\(\sqrt[5]{32}\) 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

\(\sqrt[5]{32}\) To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

**Solution**

\(\sqrt[5]{32} = 2\)

*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.

\(\sqrt{64}\)

**Answer**-
8

Practice Set B

\(\sqrt{100}\)

**Answer**-
10

Practice Set B

\(\sqrt[3]{64}\)

**Answer**-
4

Practice Set B

\(\sqrt[6]{64}\)

**Answer**-
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 \(\sqrt[7]{2187}\).

**Solution**

Display Reads | ||

Type | 2187 | 2187 |

Press | \(y^x\) | 2187 |

Type | 7 | 7 |

Press | \(1/x\) | .14285714 |

Press | = | 3 |

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

Practice Set C

Use a calculator to find the following roots.

\(\sqrt[3]{729}\)

**Answer**-
9

Practice Set C

\(\sqrt[4]{8503056}\)

**Answer**-
54

Practice Set C

\(\sqrt{53361}\)

**Answer**-
231

Practice Set C

\(\sqrt[12]{16777216}\)

**Answer**-
4

## Exercises

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

Exercise \(\PageIndex{1}\)

\(4 \cdot 4\)

**Answer**-
\(4^2\)

Exercise \(\PageIndex{2}\)

\(12 \cdot 12\)

Exercise \(\PageIndex{3}\)

\(9 \cdot 9 \cdot 9 \cdot 9\)

**Answer**-
\(9^4\)

Exercise \(\PageIndex{4}\)

\(10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10\)

Exercise \(\PageIndex{5}\)

\(826 \cdot 826 \cdot 826\)

**Answer**-
\(826^3\)

Exercise \(\PageIndex{6}\)

\(3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021\)

Exercise \(\PageIndex{7}\)

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

**Answer**-
\(6^{85}\)

Exercise \(\PageIndex{8}\)

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

Exercise \(\PageIndex{9}\)

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

**Answer**-
\(1^{3008}\)

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

Exercise \(\PageIndex{10}\)

\(5^3\)

Exercise \(\PageIndex{11}\)

\(7^4\)

**Answer**-
\(7 \cdot 7 \cdot 7 \cdot 7\)

Exercise \(\PageIndex{12}\)

\(15^2\)

Exercise \(\PageIndex{13}\)

\(117^5\)

**Answer**-
\(117 \cdot 117 \cdot 117 \cdot 117 \cdot 117\)

Exercise \(\PageIndex{14}\)

\(61^6\)

Exercise \(\PageIndex{15}\)

\(30^2\)

**Answer**-
\(30 \cdot 30\)

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 \(\PageIndex{17}\)

\(4^2\)

**Answer**-
\(4 \cdot 4 = 16\)

Exercise \(\PageIndex{18}\)

\(1^2\)

Exercise \(\PageIndex{19}\)

\(10^2\)

**Answer**-
\(10 \cdot 10 = 100\)

Exercise \(\PageIndex{20}\)

\(11^2\)

Exercise \(\PageIndex{21}\)

\(12^2\)

**Answer**-
\(12 \cdot 12 = 144\)

Exercise \(\PageIndex{22}\)

\(13^2\)

Exercise \(\PageIndex{23}\)

\(15^2\)

**Answer**-
\(15 \cdot 15 = 225\)

Exercise \(\PageIndex{24}\)

\(1^4\)

Exercise \(\PageIndex{25}\)

\(3^4\)

**Answer**-
\(3 \cdot 3 \cdot 3 \cdot 3 = 81\)

Exercise \(\PageIndex{26}\)

\(7^3\)

Exercise \(\PageIndex{27}\)

\(10^3\)

**Answer**-
\(10 \cdot 10 \cdot 10 = 1000\)

Exercise \(\PageIndex{28}\)

\(100^2\)

Exercise \(\PageIndex{29}\)

\(8^3\)

**Answer**-
\(8 \cdot 8 \cdot 8 = 512\)

Exercise \(\PageIndex{30}\)

\(5^5\)

Exercise \(\PageIndex{31}\)

\(9^3\)

**Answer**-
\(9 \cdot 9 \cdot 9 = 729\)

Exercise \(\PageIndex{32}\)

\(6^2\)

Exercise \(\PageIndex{33}\)

\(7^1\)

**Answer**-
\(7^1 = 7\)

Exercise \(\PageIndex{34}\)

\(1^{28}\)

Exercise \(\PageIndex{35}\)

\(2^7\)

**Answer**-
\(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 128\)

Exercise \(\PageIndex{36}\)

\(0^5\)

Exercise \(\PageIndex{37}\)

\(8^4\)

**Answer**-
\(8 \cdot 8 \cdot 8 \cdot 8 = 4,096\)

Exercise \(\PageIndex{38}\)

\(5^8\)

Exercise \(\PageIndex{39}\)

\(6^9\)

**Answer**-
\(6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 10,077,696\)

Exercise \(\PageIndex{40}\)

\(25^3\)

Exercise \(\PageIndex{41}\)

\(42^2\)

**Answer**-
\(42 \cdot 42 = 1,764\)

Exercise \(\PageIndex{42}\)

\(31^3\)

Exercise \(\PageIndex{43}\)

\(15^5\)

**Answer**-
\(15 \cdot 15 \cdot 15 \cdot 15 \cdot 15 = 759,375\)

Exercise \(\PageIndex{44}\)

\(2^{20}\)

Exercise \(\PageIndex{45}\)

\(816^2\)

**Answer**-
\(816 \cdot 816 = 665,856\)

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 \(\PageIndex{47}\)

\(\sqrt{16}\)

**Answer**-
4

Exercise \(\PageIndex{48}\)

\(\sqrt{36}\)

Exercise \(\PageIndex{49}\)

\(\sqrt{64}\)

**Answer**-
8

Exercise \(\PageIndex{50}\)

\(\sqrt{121}\)

Exercise \(\PageIndex{51}\)

\(\sqrt{144}\)

**Answer**-
12

Exercise \(\PageIndex{52}\)

\(\sqrt{169}\)

Exercise \(\PageIndex{53}\)

\(\sqrt{225}\)

**Answer**-
15

Exercise \(\PageIndex{54}\)

\(\sqrt[3]{27}\)

Exercise \(\PageIndex{55}\)

\(\sqrt[5]{32}\)

**Answer**-
2

Exercise \(\PageIndex{56}\)

\(\sqrt[7]{1}\)

Exercise \(\PageIndex{57}\)

\(\sqrt{400}\)

**Answer**-
20

Exercise \(\PageIndex{58}\)

\(\sqrt{900}\)

Exercise \(\PageIndex{59}\)

\(\sqrt{10,000}\)

**Answer**-
100

Exercise \(\PageIndex{60}\)

\(\sqrt{324}\)

Exercise \(\PageIndex{61}\)

\(\sqrt{3,600}\)

**Answer**-
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 \(\PageIndex{63}\)

\(\sqrt{1,156}\)

**Answer**-
34

Exercise \(\PageIndex{64}\)

\(\sqrt{46,225}\)

Exercise \(\PageIndex{65}\)

\(\sqrt{17,288,964}\)

**Answer**-
4,158

Exercise \(\PageIndex{66}\)

\(\sqrt[3]{3,375}\)

Exercise \(\PageIndex{67}\)

\(\sqrt[4]{331,776}\)

**Answer**-
24

Exercise \(\PageIndex{68}\)

\(\sqrt[8]{5,764,801}\)

Exercise \(\PageIndex{69}\)

\(\sqrt[12]{16,777,216}\)

**Answer**-
4

Exercise \(\PageIndex{70}\)

\(\sqrt[8]{16,777,216}\)

Exercise \(\PageIndex{71}\)

\(\sqrt[10]{9,765,625}\)

**Answer**-
5

Exercise \(\PageIndex{72}\)

\(\sqrt[4]{160,000}\)

Exercise \(\PageIndex{73}\)

\(\sqrt[3]{531,441}\)

**Answer**-
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.

**Answer**-
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?

**Answer**-
Yes; 0

Exercise \(\PageIndex{78}\)

Use the numbers 4 and 7 to illustrate the commutative property of multiplication.