r 2 3L - 3 Zero and Negative Exponents Zero as an Exponent: For every nonzero number a, a =1. Example 1:

(1)

Zero and Negative Exponents

Zero as an Exponent:

For every nonzero number a, a° =1. Examples:

r°

2

3L

-‘

3

Negative Exponent:

For every nonzero number a and integern, a =

Examples:

I

Example 1: Simplifying a Power Simplify.

a. 43 b. (—l.23)°

Example 2: Simplifying an Exponential Expression Simplify each expression.

b.-i

_

Example 3: Evaluating an Exponential Expression Evaluate 3m2t2 form = 2 andt=-3.

37

_

(2)

Multiplication Properties of Exponents Mu1tiplyin Powers With the Same Base Property:

For every nonzero number a and integers m and n, am _a” ₌ _a_m_.

Examples: 5)_

5

2TLi

51

2’

Exam pie 1: Multiplying Powers

Rewrite each expression using each base only once.

a. ii4 •ii3 b. 52 .52

(

2-2

[1

‘

5

Example 2: Multiplying Powers in an Algebraic Expression Simplify each expression.

a. 2n5 3n2 b. 5x2y4 •3x8

4

°LN

Raising a Power to a Power Property:

For every nonzero number a and integers m and n, (atm)” =atm”.

Examples: _2..\)3

5-

(—2’)

Example 3: Simplifying a Power Raised to a Power Simplify (x3)6

(3)

Example4: Simplifying an Expression With Powers

Simplify c’(c3)2.

Example 5: Simplifying a Product Raised to a Power

Write the expression that represents the area of the square.

2x

tL

(axjz

2

Example 6: Simplifying a Product Raised to a Power Simplify (x2)2(3xy2)4.

N,

cHI1f

,J:71

Li

Raising a Product to a Power Property:

For every nonzero number a and b and integer n, (ab)’7 = a’.

Examples:

(3\)a.

32. 2

92

(4)

Division Properties of Exponents

Example 1: Simplifying an Algebraic Expression Simplify each expression.

a.

-2

c1d3

ft.

L

Example 2: Raising a Quotient to a Power Dividing Powers With the Same Base Property:

For every nonzero number a and integers mand n, = a’3.

Example:

2

r

1

b

Cd

1

Raising a Ouotient to a Power Property:

For every

nonzero number a and

band integer n(J =

Example:

(

/

(4N3

Which expression is equivalent to I —i- ?

x)

H

3

(L

(5)

-Example_{3: Simplifying an Exponential Expression}

Simplify each expression.

a. H— —

(6)

Section 7.1 Roots and Radical Expressions Since 52 =25, 5 is a ____________ root of 25. Since 53 = 125, 5 is a ______________ root of 125.

Since 54 =625,5isaC(+k rooto

Since 55 =3125,5isa

_________

rootof

3j

Definition of thenth Root:

For any real numbers a

and

b,

and any

positive integer n, if a’ =b, then a is an nth

root of b. -

9

oc

c*d

— O

4-

rcc*s

)-I(hC5flOftoi

-‘

khQ C

4Z

j

Summary of the possible real roots of a real number.

Numberof Rea’ nth

NLInbef’

of ReI ,,th

Type ofNimiber Roots Whenns Even ROGtSWeii IFJs Odd

-:JciF(\)

c

QOc

h

ñ

Example 1: Finding All Real Roots Find all the real roots.

a. The cube roots of 8, -1000, and

t\

I

-DC

hQ1\cc4

Iäo

0

\‘\

hQ\’Q_I

OO&

16 b. The fourth roots of 1, -0.0001, and —.

L

‘jiI.t

c{th rooi-

od

-

VH and

_(-r)Z

I

V

1T)

hCJL QJJ

(TJ3

S

_1JTD1O

b

1O

V

t

&--- , -I 1 ___

5

(7)

Radical Sign:

\,cJ —ko

\r$&CCk(

Radicand:

—[-

cctr rdr

*

tijç oJ

Index:

Th°

-.

PrincipalRoot:

-JQ

n c

n’JU)cr

\CkS

+o oc*.

TYL i

nc

4 rc

O

6

ThQ

Lth

rc’ct

_Q$

T

Example 2: Finding Roots Find each real-number root.

C. Jx’y

c@oJ iu&br

3

a

_codcO

b. J—ioo

‘S

Sc)c\m

a.Vi

-;

Example 3: Simplifying Radical Expressions Simplify each radical expression.

a.

b. Ja3b6

(8)

Section

7.2

Multiplyin2 and Dividing Radical Expressions

MultipIyin Radical Expressions Property:

If and ‘[ are real numbers, then

.

=

Example

1:

Multiplying Radicals Multiply. Simplify if possible.

-\

=

.

pap+j cbQS nc*

cppk scc

\

-

S

rob-

a

QcJ

nuiDg.

Example 2: Simplifying Radical Expressions

Simplify each expression. Assume that all variables are positive. Then absolute value symbols are never needed in the simplified expression.

a.

\J

__

b. kI8On

J

10

z

)ion

Dividing Radical Expressions Property:

(9)

Example 4: Dividing Radicals

Divide and simplify. Assume that allvariables arepositive.

a.

b

3

___

Rationalize the Denominator

f)J

ç-

-\-\Q-

áQf1ofl

()Q

O

\Dd

cc.

Example 5: Rationalizing the Denominator

ncO1

Rationalize the denominator of each expression. Assume that all variablesarepositive.

a

•J

\{3

_1çI

-

5X\/

b c. 3J_

V

3x ‘I

4D

—

.L’E

-c

L_

(10)

Section 7.3

-

Binomial Radical Expressions

Like Radicals:

ea\ ftSS

[CC

Example!: Adig ubtingdial

-W’&... bxiQ

Add or subtract if possible.

a.

5k1—3&

b.

4Ii+5J

rcC-Example 2: Simplifying Before Adding and Subtracting Simplify 6-lu +

4J

—

3-In.

J9Y

-33

4l_

+

-L:

Example

3: Multiplying Binomial Radical Expressions Multiply (3+ 2J)(2+

4J).

Pci :

t

4

(11)

Example 4: Multiplying Conjugates Multiply (2+ J)(2—

Fo

IL

cc

Q

T

DrQnc c

- \

(cbbcb2

()z

Example 5: Rationalizing Binomial Radical Denominators

3+

Rationalize the denominator of

-(3

4

I

-‘-5

-Lj

(12)

Section 7.4

Rational Exponents

Rational Exponent:

-

CO

1sScor.

VL

Example 1: Simplifying Expressions With Rational Exponents Simplify each expression.

a. 125

b. 5 5}

\)

5

1o.1oo =

C)

Definition of Rational Exponents: -

TL

(Th(

If the nth root of a is a real number and m is an integer, then

a = _{and a} ₌

(

‘J

OCrdk

Example

2: Converting to and From Radical Form

a. Write the exponential expressions x and y2.5 in radical form.

b. Write the radical expressions

J1

and (J)2 in exponential form.

b

2

(13)

Example 4: Simplifying Numbers With Rational Exponents

ExampleS: Writing Expressions in Simplest Form Write (16y8) in simplest form.

-Summary of Properties of Rational Exponents:

Let m

and

n represent rational numbers. Assume that no denominator equals 0.

Example

]‘Lg

4

( )2.. _L3 -

5

Property a 11 ’ a am+n (am)n =am (ab) 11 =a11n11 _n a a” m = a 11 -1) L 9Vi f \11 (a a [L’\3 “ b 11

ai

3

Simplify each number.

a. (—32)

_33/5

\j(33)

3

. = b. 435 L L4 N I

1(s”I

\

1

J4/

1

\j

—_i-—

Ni 1 )

3

(14)

.-Section

7.5

Solve Square Root and Other Radical Equations

Radical Equation:

ç-

rn

\xI) @ ‘\Jc

QE*

0

c

Solving a Radical Equation.

sc tcU

+hp

rod

_cc] _c

c+-

+

cjxthc

-

thSQ

boW’

Qa\ O

-Exampie

1: Solving Square Root Equations

Solve 2+iJ3x—2=6.

\3K-

H

CHY

-i-•

-Example

2: Solving Radical Equations With Rational Exponents

____

X-

53

tb

x

-Solve 2(x—2) =50.

(x

‘A

-or Sc&

c$

+i4

CbQC.

0

cj9

r’3

3Q

::

50

:

50

• (

50

53DQ

5Q.

V

.‘

-.

1

‘F I’ cNc

x-.--.

.(ES

I

x-\) 9

(15)

Extraneous Solutions:

A tuEcn

d

cn

c± cn

•i

Qd

om

Example 3: Checking For Extraneous Solutions

32jj

f5

-4-3

_o

Example 4: Solving Equations With Two RationalExponents Solve (2x+ 1)0.5 —

(3x

+

4)0.25

0. Check for extraneous solutions.

Q3

oc

/

“I) Solve -Jx —3 +5 = x. Check

for

extraneous solutions.

5K

-5

A

Al)

(7S

(i-’Xx

\rTE ÷31

J+

5

7

1 7 ,/

\J

÷5

7 .3

—H

ChQch

(./)

+

O05

/3/

.

17 ÷ \ -.

(%)°

/ 5iEz --L (i:) .7 --

(3(/)

/c -1- L

)

(3KH)

c

(x

i)a

(±i)

_z(*t)

q

4

q+(

3x4

4xx

-3

D(S

o

1

.

--,

q

--So

I

(3

(E:)

0

-1

f

\Q.5

v-I)

2

±0

rai