§ 7.2

Radical Expressions and

Functions

Tobey & Slater, Intermediate Algebra, 5e - Slide #2

Square Roots

The square root of a number is a value that when multiplied by itself is equal to the original number.

The positive square root is called the principal square root.

Definition of Square Root

If x is a nonnegative real number, then is the nonnegative

(or principal) square root of x; in other words,

x

2 .x x

4

Radical sign

2

Radicand

n xRadical expression

Index

When no number for n appears, 2 is the index.

Tobey & Slater, Intermediate Algebra, 5e - Slide #3

Higher Order Roots

Definition of Higher Order Roots1. If x is a nonnegative real number, then is a nonnegative

nth root and has the property that

2. If x is a negative real number, then

n x

.nn x x

a.) , when is an .nn x x n odd integer

b.) is a real number, when is an .nn x not n even integer

Example: Find the root of 3 1000.

3 1000 3 3( 10) 10

“Cube root”

Tobey & Slater, Intermediate Algebra, 5e - Slide #4

Square Root Functions

The square root function f(x) has a domain of all real numbers x that are greater than or equal to 0.

Example: Find f(3) for the function and find the domain.

3 5( ) , f x x

( ) = -3-3 -( 53) -f

9 5

42

To find the domain, we know that 3x – 5 must be nonnegative.3 5 0 x

3 5 x53

x

The domain is all real numbers

x where 53.x

Tobey & Slater, Intermediate Algebra, 5e - Slide #5

Expressions with Rational Exponents

If n is a positive integer and x is a nonnegative real number, then1/ .n nx x

Example: Change w3/4 to radical form.

43/4 3w w

Example: Change (3abc)2/5 to radical form.

2/ 253 3abc abc

5 2 2 29 a b c

Tobey & Slater, Intermediate Algebra, 5e - Slide #6

Evaluating Higher Order Radicals

For all real numbers x (including negative real numbers), then2 when is an positve integer,n x x n even

Example: Simplify.

2 when is an positve integer.n x x n odd

4 204a.) x y 3 3b.) -27a

4 20 54 x y xy 3 6 227 3 a a