How do I use properties of exponents to simplify expressions involving powers?

• How do I use properties of exponents to simplify expressions involving powers?

4.1 Use Properties of ExponentsProperties of ExponentsLet a and b be real numbers and let m and n be integers.

Product of Powers Property ______aaa nm nm

Power of a Powers Property ______aanm mn

Power of a Product Property ______baab m m m

Negative Exponents Property 0 ,______ aa mma

1

Zero Exponent Property 0 __,0 aa 1

Quotient of Powers Property 0 ,______ aaa

an

mnm

Power of a Quotient Property 0 ,_________

ab

am

m

m

b

a

4.1 Use Properties of Exponents

Example 1 Evaluate a numerical expressionEvaluate a numerical expression

326 a. ____6 32 __6 6 ______ 656,46

3

5

4

4 b. ____4 35 __4 2 ___16

47 c.

_________

47

1_________

2401

1

4.1 Use Properties of ExponentsCheckpoint. Evaluate or simplify the expression.Checkpoint. Evaluate or simplify the expression.

1

2

5

8

8 1.

58

58

2838

1

512

1

28

4.1 Use Properties of Exponents

Example 2 Use scientific notation in real lifeUse scientific notation in real lifeIceland covers about 1.03 X 105 square kilometers and has a population of approximately 2.94 X 105 people. About how many people are there per square kilometer?Solution

_______________________________area Land

Population

51094.2 51003.1

Divide population by land area.

Quotient of powers property

_______________________________

03.1

94.2 010

Use a calculator__________ 85.2 010Zero exponent property_____ 85.2

There are about ___ people per square kilometer.3

4.1 Use Properties of Exponents

Example 3 Simplify expressionsSimplify expressions

815

325

a.yx

yx815

yx Power of a product

property 35x 32y

815

yx Power of a power

property

15x6y

_______Quotient of powers property

1515x 86y

_____ Simplify exponents0x 2y

_____ Zero exponent property2y

___________

Negative exponent property2

1

y

4.1 Use Properties of Exponents

Example 3 Simplify expressionsSimplify expressions2

2

4

b.

b

a

Power of a quotient property

24a

22b

Power of a power

property

8a4b

Negative exponent property

148ba

4.1 Use Properties of ExponentsCheckpoint. Simplify or evaluate the expression. Tell Checkpoint. Simplify or evaluate the expression. Tell which properties of exponents you used.which properties of exponents you used.

23 107104 2.

74 110Product of powers property

11028280

2108.2

4.1 Use Properties of ExponentsCheckpoint. Simplify or evaluate the expression. Tell Checkpoint. Simplify or evaluate the expression. Tell which properties of exponents you used.which properties of exponents you used.

3

5

22 3.

ba

ba

Power of a quotient/product property 2 3

32a 3b

35a 3b Power of a power property

8

6a 3b15a 3b

Quotient of powers property

8

9a

4.1 Use Properties of Exponents

Example 4 Compare real-life volumesCompare real-life volumesBeach Ball The radius of a beach ball is about 5.6 times greater than the radius of a baseball. How many times as great as the baseball’s volume is the beach ball’s volume?Let r represent the radius of the baseball

3

3

34

34

volumesBaseball'

volumesball'Beach

r

r6.5 The volume of a sphere is

Power of a Product property property

3 34

34

r

336.5 r

. 3

4 3r

4.1 Use Properties of Exponents

Example 4 Compare real-life volumesCompare real-life volumesBeach Ball The radius of a beach ball is about 5.6 times greater than the radius of a baseball. How many times as great as the baseball’s volume is the beach ball’s volume?Let r represent the radius of the baseball

_______ 036.5 r Quotient of Powers

Zero Exponent Property____ 36.5Approximate power____176

The beach ball’s volume is about ____ times as great as the baseball’s volume.

176

4.1 Use Properties of ExponentsCheckpoint. Complete the following exercise.Checkpoint. Complete the following exercise.4. Rework Example 4 where the radius of a

beach ball is about 6 times the radius of a baseball.

3

3

34

34

r

r63

336

r

r 36 216

216 times as great

4.1 Use Properties of Exponents

Pg. 117, 4.1 #1-42