Integer Exponents

8.EE.1

Objective - To solve problems involving integer exponents.

Exponential Form

Exponential vs. Expanded Form

Expanded Form:

7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7 ∙ 7

Exponential Form:

713

Try these

Expanded Form:

(–3) ∙ (–3) ∙ (–3) ∙ (–3) ∙ (–3) ∙ (–3) ∙ (–3)

Exponential Form:

(–3)7

Expanded Form:

(⅔) ∙ (⅔) ∙ (⅔) ∙ (⅔)

Exponential Form:

(⅔)4

NOTE

a = a1

You should always write the invisible 1 to help you with problems involving exponents.

Bases and Exponents

x x ∙ x ∙ x=

exponent

base

3

The BASE tells us what is being multiplied.

The EXPONENT tells us how many times to multiply the base.

Identifying the Base & Exponent

25 The base is 5. The exponent is 2.

25 The base is 5. The exponent is 2.

2)5( The base is - 5. The exponent is 2.

5)3( x The base is (x+3). The exponent is 5.

Evaluating Exponents

25 The base is 5. The exponent is 2.

2)5( The base is - 5. The exponent is 2.

)55(52 25

)5)(5()5( 2

25

Exponent VocabularyWhen we raise something to the second

power, we use the word

SQUARED

72

Seven squared

When we raise something to the third power, we use the word

CUBED

53

Five cubed

Simple Rule

• If the base is negative:– An even exponent means a positive answer– An odd exponent means a negative answer

Example:

2)3( 9 3)3( 27

Multiplication Rules

Product of Powers Property

When you MULTIPLY quantitieswith the same base

ADD the exponents.

0 where, is rule general The aaaa nmnm

Division Rules

Quotient of Powers Property

When you DIVIDE quantities with the same base

SUBTRACT the exponents.

Always top exponent minus bottom exponent!

0 where, is rule general The aaaa nmnm

0 where,

as written be alsocan This

aaa

a nmn

m

10except , numbers real allFor 0 , a a a

Zero Exponent Property

If a division results in the complete cancellation of all factors then the answer

is 1.

07

0r

015

0

7

2

1

1

1

−1

TRY THESE

Power of a Power Rule

Example:

baba xx

23n 23n 6n

33 nn 33n

)( nnn

6n

)( nnn 6n

Power of a Product Rule(Distributive Property of Exponents over Multiplication)

Example:

nnnn yxxy AA

2535 ba

252325 ba 10625 ba23a 25b

Negative Exponent Property

For all real numbers a, a ≠ 0, and n is an integer:

nn

aa

1

Working with Negative Exponents

When you see a negative exponent,

think FRACTION!

If no fraction exists,

create a fraction,

by putting what you have over 1!

Examples:4

14 1

55 1

xx

Fractions As Bases

If you have a fraction as the base,and there is a negative exponent:

FLIP THE FRACTION!

Example: 2

5

2

2

2

5

2

5

2

5

4

25

Simplify.

1)

2)

3)

4)

5)

6)

13

15

24

26

14

5

22

3

1

3

1

5

2

1

4

1

16

2

1

6

1

36

5

4

23

2

9

4

Follow the Pattern!3323130313

23

33

= 3 3 3= 3 3= 3

= 1

=1

3

=1

3

1

3

=1

3

1

3

1

3

33

3

3

3

3

Fractional Exponents

CP Classes Only

Negative Exponents in DenominatorEvaluate.

2

3

x 23 x 2

3 1

1 x

23 x

1 1 23x

2

3

x 23x

Simplify.

1)

2)

3)

4)

4

1

3

3

x

y

2

1

5a

3 4

m

7x y

43 81

3xy

25a 225a

3 4mx y

7

Examples:

239

1

3

12

7x 7

1

x

2

9

2 y29

2

y

cb

a3

5

4

ca

b5

3

4

31

4

4

2 zy

x

2

34 yzx

83

17

20

15

ts

rq

8

37

4

3

rt

sq

Simplify.1)

2)

3)

4 25 5

2 1(8 )

2 3 5(7 ) 7

4)

5)

6)

2 43 3 3

3 22 5

3 2(4 4)

4 25

25 25

2 ( 1)8

28 1

64

6 57 7 17 1

7

2 4 13

13 1 3

18

25

8

25

3 1 2(4 )

2 ( 2 )4 44 256

6 4

4

a b

2

3

m

Simplify.1)

2)

3)

23m

26xy

2(6xy)

4)

5)

6)

4 4y y

3 3(2xy )

3 2 4(2a ) (b)

2

13

m

2

16x

y

2

6x

y

2

1

(6xy) 2 2

1 36x y

44

1y

y 1

3 3

1

(2xy ) 3 9

1 8x y

2

3 4

2 1

a b

1

2x = x or 2 x

1

3x = 3 x

1

4x = 4 x

A fractional exponent is a root of the base.

Simplify.1

381)

2)

3)

4)

5)

6)

1

216

1

416

1

481

1

532

1

3125

=

=

=

=

=

=

3 8 = 2

16 = 4

4 16 = 2

4 81 = 3

5 32 = 2

3 125 = 5

Roots of Negative Numbers1

2( 16) = 16 = No Real Root

1

3( 27) = 3 27 =3 3 3 27

3

1

5( 32) = 5 32 =

( 2 )( 2 )( 2 )( 2 )( 2 ) 32 2

1

4( 81) = 4 81 = No Real Root( 3)( 3)( 3)( 3) 81

1)

2)

3)

4)

–4 ● – 4 ≠ – 16

Write using a root. Simplify.1

2( 9) = 9 =

1

3( 8) = 3 8 = 2

1

3( 125) = 3 125 = 5

1

4( 16) = 4 16 =

1)

2)

3)

4)

No Real Root

No Real Root

Write using a root. Simplify.1

5( 32) = 5 32 =

1

6( 64) = 6 64 =

1

3( 1000) = 3 1000 = 10

1

7( 128) = 7 128 =

5)

6)

7)

8)

2

No Real Root

2

Rule of Rational Exponents1

nx = n xor

a

bx = b ax = ab x

a

bx

power (exponent)

root2

38 =1

238

=1

23(8 ) = 2(2) = 42

38 =1

238

=

12 3(8 ) =

1

3(64) = 4

Root or power first?Doesn’t matter!

a

bx

power (exponent)

root