7.1 Radical Expressions

Objective 1: Find principal square roots of numbers

A square root of a number a is a number c such that

Examples: 1. 25 has a square root of 5 because 2. 25 has a square root of -5 because3. -16 does not have a real-number square root

because there is no real number c such that

2c a

25 252( 5) 25

2 16c

Theorem 7-1

-Every positive real number has two real-number square roots.

-The number 0 has just one square root, 0 itself.-Negative numbers do not have real-number

square roots.Ex. Find the two square roots of 64.The square roots are 8 and -8.

Try This

Find the square roots of each number.

1. 92. 363. 1214. 05. -49

Definition

The principal square root of a nonnegative number is its nonnegative square root. The symbol represents the principal square root of a. the negative square root of a is written .

Ex. Simplify.1. 2. 3. 4.

a

a

25 64 25

640.0049

Try This

Simplify.6. 9.

7. 10.

8. 11.

1

36

81

100

0.0064

0.0081

144

Definition

The symbol is a radical sign. An expression written with a radical sign is a radical expression. The expression written under the radical sign is the radicand.

Theorem 7-2

For any real number a, . The principal (nonnegative) square root of is the absolute value of a.

Ex. 1. 3.

2. 4.

2a a2a

2( 16) 16 16

2( 1) 1x x

2(3 ) 3 3b b b

2 28 16 ( 4) 4x x x x

Try This

12. 13.

14. 15.

2( 24) 2(5 )y

216y 2( 7)x

Objective 2: Find odd and even kth roots

The number c is the cube root of a if .• 2 is the cube root of 8 because .• -5 is the cube root of -125 because

Ex. Simplify.1. 2. 3. 4.

3c a32 2 2 2 8

3( 5) ( 5)( 5)( 5) 125

3 8 3 27 364

125

33 8y

Try This

Simplify.16.

17.

18.

33 8y

33 27y

3343

64

Rewrite using exponential notation

1.

2.

3.

77c b c b

66 100 100y y

11 22t

Try This

19.

20.

21.

5b a

12 63c

9 16a n

The number k in is called the index. If k is an odd number, we say that we are finding an odd root.

Examples. Find the following.1. 2. 3. 4.

k

5 32 5 32

7 7x99 ( 1)x

Try This

Find the following.22.

23.

24.

25.

5 243

5 243

5 532x

77 (3 2)x

Theorem 7-3

For any real number a, the following statements are true.

A. When k is even.

B. When k is odd.

k ka a

k ka a

Try This

Find the following.

26. 29.

27. 30.

28. 31.

4 81

4 81

4 81

44 16( 2)x

88 ( 3)x

3 27x