Ch 8 - Rational & Radical Functions

Simplifying Radical Expressions

Product Property of Radicals:For any real numbers a and b,

then

baab

Simplify.

330. aA

75454. zyxB

32 4010. xyyxC

22 612. mnnmD

Quotient Property of RadicalsFor any real numbers a and b,

and b ≠ 0,

if all roots are defined.

To rationalize the denominator, you must multiply the numerator and denominator by a quantity so that the radicand has an exact root.

b

a

b

a

Simplify.

4

5.A

22

820.B

32

6.C

32

6.

knD

Two radical expressions are called like radical expressions if the radicands are alike.

dcba Conjugates are binomials in the form

and

where a, b, c, and d are rational numbers. The product of conjugates is always a rational number.

dcba

Simplify.

26723210. A

15024365. B

72547254. C

56

521.

D

nth roots

The nth root of a real number a can be written as the radical expression , where n is the index of the radical and a is the radicand.

n a

Numbers and Types of Real Roots

Case Roots Example

Odd Index 1 real root The real 3rd root of 8 is 2.The real 3rd root of -8 is -2.

Even Index, Positive Radicand

2 real roots The real 4th roots of 16 are ±2.

Even Index,Negative Radicand

0 real roots -16 has no real 4th roots.

Radicand of 0 1 root of 0 The 3rd root of 0 is 0.

Find all real roots.A.Sixth roots of 64

B.Cube roots of -216

C.Fourth roots of -1024

Properties of nth Roots

Product Property of Roots

nnn baab

Quotient Property of Roots n

n

n

b

a

b

a

Simplify each expression. Assume that all variables are positive.4 1281. xA 4

8

5

16.

xB

A rational exponent is an exponent that can be expressed as m/n, where m and n are integers and n ≠ 0.

The exponent 1/n indicates the nth root.

nn aa 1

n mmnn

m

aaa The exponent m/n

indicates the nth root raised to the mth power.

Write each expression by using rational exponents.

8 413.A5 153.B

32

125. C 53

32. D

Write each expression in radical form and simplify.

Properties of Rational Exponents

Product of Powers Property

Quotient of Powers Property

Power of a Power Property

Power of a Product Property

Power of a Quotient Property

nmn

m

aa

a nmnm aaa

m

mm

b

a

b

a

mmm baab mnnm aa

Simplify each expression.

9

11

9

7

77. A4

5

4

3

16

16.B