7. Roots and Radical Expressions

In this chapter, you will learn:

• What a polynomial is

• Add/subtract/multiply/divide polynomials

• Simplify radicals, exponents

• Solving equations with exponents and radicals

• Complex numbers

• Conjugates

What is a monomial?

An expression that is a number, that may or may not include a variable.

x4

2x2xy10

MONOMIALS NOT MONOMIALS

x

48x

x518

Real Roots• Real roots are the

possible solutions to a number, raised to a power.

powerfourth the to16 of roots

are 2 negative and 2both so

16)2(2 44

5?) beit t can'(why

125- ofroot real

possibleonly theis 5- so

12553

power

second the x toof roots

arex - andboth x so

2

xxor

xxx

Vocabulary and Properties

n aindex

Radical sign

radicand

How to find the root (other than a square

root), using a graphing calculator 1. Input the root you are going to

take (for example, if you are taking the third root of a number, start with the 3).

2.Press MATH and select option 5

3. Enter the value you are taking the root of.

Ex:

x

4 81 4 MATH 5 81 ENTER

3

Practice: Find each root

3 10648

5 16807

Solutions: 22, 7, and

ERR: NONREAL ANS

4 16Let’s take a closer look at this answer

Properties and Notation:

aan When n is an even number

Why? We want to make sure that the root is always positive when the index is an even number

positive. stays

it sure make to valueabsolute theuse Weforwards. and

backwards trueisit that so indexeseven with

positive a always is that x sure make want to we

525)5(,5 if BUT

5255,5 IF

:itat look y toanother wa sHere'

22

22

so

xxx

xxx

24 42444 84 2)()2(16

:

yxyxyx

ex

Note: Absolute value symbols ensure that the root is positive when x is negative. They are not needed for y because y2 is never negative.

23 3233 63 )(

:

yxyxyx

ex

Absolute value symbols must not be used here. If x is negative, then the radicand is negative and the root must also be negative.

Notice that the index is an odd number here . . .

Let’s try some

3 627 yx

Simplify each expression. Use the absolute value symbols when needed.

424 yx

Solutions

3 627 yx

Simplify each expression. Use the absolute value symbols when needed.

424 yx

Properties of Exponents – let’s review . . .

NEGATIVE EXPONENTRULE

nn

a

1a

2525

1

PRODUCT OR POWERRULE

2010 22 nmnm aaa

302

HAVE TO HAVE THESAME BASE

QUOTIENT OF POWERRULE

4

10

3

3 63

bab

a

xx

x HAVE TO HAVE THESAME BASE

POWER OF POWERRULE

(x4)³ 34x 12x

mnnm a)a(

POWER OF PRODUCTRULE

mnnnm ba)ab(

(2x4)⁵ 545 x2 20x32

POWER OF A QUOTIENTRULE

a

b

a

b

n n

n

FHGIKJ

35

y

FHGIKJ

35

5y

POWER OF QUOTIENT 2RULE

2

y

3

2

2

y

3

2

2

3

y

9

2y

a

aa

x

y

y

x

Fractional Exponents (Powers and Roots)

xyy xy

x

aaa )(“Power”

“Root”

RADICAL TO EXPONENTRULE

251 2/

a a1 2/

25 5

a an n1/

RATIONAL EXPONENTRULE

mnn mn

m

aaa

4

3

16 34 16 32 8