+

Using Properties of ExponentsEQ: How do we use properties of exponents?

M2 Unit 5a: Day 1

Wednesday, April 19, 2023

+Properties of ExponentsLet a and b be real numbers and let m and n be integers.

Product of Powers Property

Power of a Power Property

Power of a Product Property

Zero Exponent Property

Quotient of Powers Property

Power of Quotient Property

Negative Exponent Property

am an amn

(am)namn

(ab)mambm

a0 1, a0

am

anam n, a0

ab

m

am

bm, b01 , 0m ma aa- = ¹

+Product of Powers:

Examples:

a.

b.

(x4y5)(xy3)

To multiply powers with the same base, add the exponents and keep the common base.

x6 x3

x9

x5y8

+

Your turn:1.

2.

3.

b4 b

(2ab2c)(3a3b)

x 2 yx 2 y2

Keep common

bases

Add the

exponents

b5

6a4b3c

x 4 y3

+Power of a Power:

Examples:

a.

b.

(53)2

(x 6)3

To raise a power to a power, keep the base and multiply the exponents.

56

x18

15625

+

Your turn:

1.

2.

3.

(s7)2

(42)3

(( 2)2)3

KeepThe

Base

MultiplyThe

Exponents

s14

46

4096

( 2)6

64

+Power of a Product:

Examples:

a.

b.

(3a)2

(5x 2 yz4)2

To raise a product to a power, raise each factor to the power.

This should remind you of the distributive property. Remember to “distribute” the exponent to each base, numeric and variable! Warning: No distributing over addition!!!

32a2

9a2Simplify any

numeric bases raised to

exponents

52 x 4 y2z8

25x 4 y2z8

+

Your turn:

1.

2.

3.

Raise EachFactor ToThe Power

It’s Like The Distributive

Property

(x 5 y2)3

(3a2b)4

(7m7)2

x15y6

34a8b4

81a8b4

72 m14

49m14

+Zero Exponent:

Examples:

a.

b.

3xy0

(55abc2)0

Any number raised to the zero power is equal to “1”.

3x 1

3x

1

+

Your turn:

1.

2.

3.

(2xy4)0

457030a2b

(2x 2 y0)0

AnythingRaised To The Zero

Power

=1

1

a2b

1

+Quotient of Powers:

Examples:

a.

b.

15x 3 y6

5xy3

18b7c6

81b10c2

To divide powers with the same base, subtract the exponents and keep the common base.Also, remember that common factors cancel out!

3x 31y6 3

3x 2 y3

2c6 2

9b10 7

2c4

9b3

Keep the base where the

exponent is larger

+

Your turn:

1.

2.

3.

6x 5 y2

xy

20a2b2

50a

Common Factors

Cancel Out

With Common

BasesSubtract

Exponents

2mn5

36m2n3

6x 51y21

2a21b2

5

n5 3

18m21

n2

18m

2ab2

5

6x 4 y

+Power of Quotient:

Examples:

a.

b.

2xy2

3

m2np

2

To raise a quotient to a power, raise the numerator and the denominator to the power.

23 x 3

y6

m4n2

p2

8x 3

y6

+

Your turn:

1.

2.

3.

xy3

5z2

4

152x

3

xyzabc

3

Raise the numerator

and denominator to the power

Simplify any factors with

numeric bases

x 4 y12

54 z8

x 4 y12

625z8

153

23 x 3

33758x 3

x 3 y3z3

a3b3c3

+

You may want to think of it this way: unhappy ( ) exponents will become happy ( ) by having the base/exponent pair “switch floors”!

Example:

A.) B.)2x y-32-

Negative Exponent Property:

2

y

x=3

1

2=

1

8=

+You try:

1.

2.

3.

4.

y 5

2a 2b4

13

x 3y

a 1b 2

1

y5

2b4

a2

13x3

y

1

ab2

+Simplifying Exponential Expressions

To simplify exponential expressions completely:

Get rid of parentheses

Get rid of negative exponents

Make sure there is only one of each base

Cancel out common factors

+Simplify the exponential expression.Examples:

A.

8p3q9

2p2q11

B.

3a 3

9b 2

4xy3z2 2

D.

mn3

p

5

mu

ltistep

C.

= 2

4pq

=2

33ba

= 2 6 416x y z

=5 15

5

m np

+

1.

2.

3.

4.

You try:- -3 4 2( )k m

æ ö÷ç ÷ç ÷çè ø

0

2

3xz

-

- -

0 6

3 7

164

q rq r

--

-

æ ö÷ç ÷ç ÷ç ÷è ø

27 2

33x y

y

mu

ltistep

=6

8

km

=1

= 34q r

= 14 2

9x y

+Simplify the exponential expression.A.

B.

C.

D.

- 3 5 8k k k

--æ ö÷ç ÷ç ÷ç ÷è ø

23

5

bc

-

-

5 4

4

93s tt

- -

7 4

2 1 2( )y z

z z

mu

ltistep

= 10k

= 6 10b c

= 53s

= 7y

+You Try:

5.

6.

7.

-- 5 4 2( 3 )p q

-æ ö÷ç ÷ç ÷ç ÷è ø

14hggk

- ×1 3 2 4( )x x y

mu

ltistep

=8

10

9qp

= 3

khg

= 8 8x y