Radical Expressions

MATH 018

Combined Algebra

S. Rook

2

Overview

• Section 10.1 in the textbook:– Finding square roots– Finding cube roots– Finding nth roots

3

Finding Square Roots

44

Finding Square Roots

• Should be a review for numbers: means “what number multiplied by itself gives you a”?

What is the value of ?

• What about the square root of a negative number?– Suppose we want to evaluate

• The square root of a negative number does NOT exist in the real number system because the product of two negatives is positive!

a

4

100

55

Finding Square Roots (Continued)

• Not any more difficult for variables:– Consider evaluating

– What times itself will yield x4?

– Can also see by expanding x4

• Thus if a is divisible by 2• You will find it beneficial to memorize AT LEAST the

first ten perfect squares

4x

2aa xx

Finding Square Roots (Example)

Ex 1: Evaluate in the REAL number system if possible:

a) d)

b)

c) 6

64

49

6a

42b

7

Finding Cube Roots

88

Finding Cube Roots

• Should be a review for numbers means “what number multiplied by itself three times gives you a”?What is the value of ?

• What about the cube root of a negative number?– Suppose we wish to evaluate

• The cube root of a negative number EXISTS in the real number system because the product of three negatives is negative

3 a

3 8

3 64

99

Finding Cube Roots (Continued)

• Not any more difficult for variables:– Consider evaluating– What times itself will three times yields x9?– Can also see by expanding x9

• Thus if a is divisible by 3• You will find it beneficial to memorize AT

LEAST the first five perfect cubes

33 aa xx

3 9x

Finding Cube Roots (Example)

Ex 2: Evaluate in the REAL number system if possible:

a) d)

b)

c) 10

3 64

3 64

3 6a

3 42b

11

Finding nth Roots

1212

Finding nth Roots

• Should be a review for numbers means “what number multiplied by itself n times gives you a”?

How would we evaluate ?

How would we evaluate ?

• What about the nth root of a negative number?– How would we evaluate ?– How would we evaluate ?

n a

5 32

4 81

4 165 1

13

Finding nth Roots (Continued)

– Can extend this to the general case:• The product of an even number of negatives is

positive– Therefore, the even root of a negative

number does NOT exist in the real number system

• The product of an odd number of negatives is negative

– Therefore, the odd root of a negative number DOES exist in the real number system

Finding nth Roots (Example)

Ex 3: Evaluate in the REAL number system if possible:

a) d)

b)

c) 14

5 243

4 256

6 12a

5 50b

15

Summary

• After studying these slides, you should know how to do the following:– Find square roots– Find cube roots– Find nth roots

• Additional Practice– See the list of suggested problems for 10.1

• Next lesson– Rational Exponents (Section 10.2)