ExponentsExponents

35Power

base

exponent

3 3 means that is the exponential

form of t

Example:

he number

125 5 5

.125

53 means 3 factors of 5 or 5 x 5 x 5

The Laws of Exponents:The Laws of Exponents:

#1: Exponential form: The exponent of a power indicates how many times the base multiplies itself.

n

n times

x x x x x x x x

3Example: 5 5 5 5

n factors of x

#2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!

m n m nx x x

So, I get it! When you multiply Powers, you add the exponents!

512

2222 93636

#3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!

mm n m n

n

xx x x

x

So, I get it!

When you divide Powers, you subtract the exponents!

16

222

2 4262

6

Try these:

22 33.1

42 55.2

25.3 aa

72 42.4 ss

32 )3()3(.5

3742.6 tsts

4

12

.7s

s

5

9

3

3.8

44

812

.9ts

ts

54

85

4

36.10

ba

ba

22 33.1 42 55.2 25.3 aa

72 42.4 ss

32 )3()3(.5

3742.6 tsts

8133 422

725 aa

972 842 ss

SOLUTIONS

642 55

243)3()3( 532

793472 tsts

4

12

.7s

s

5

9

3

3.8

44

812

.9ts

ts

54

85

4

36.10

ba

ba

SOLUTIONS

8412 ss

8133 459

4848412 tsts

35845 9436 abba

#4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents!

nm mnx x

So, when I take a Power to a power, I multiply the exponents

52323 55)5(

#5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.

n n nxy x y So, when I take a Power of a Product, I apply the exponent to all factors of the product.

222)( baab

#6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent.

n n

n

x x

y y

So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.

81

16

3

2

3

24

44

Try these:

523.1

43.2 a

322.3 a

23522.4 ba

22 )3(.5 a

342.6 ts

5

.7t

s

2

5

9

3

3.8

2

4

8

.9rt

st

2

54

85

4

36.10

ba

ba

523.1

43.2 a

322.3 a

23522.4 ba

22 )3(.5 a

342.6 ts

SOLUTIONS

10312a

6323 82 aa

6106104232522 1622 bababa

4222 93 aa

1263432 tsts

5

.7t

s

2

5

9

3

3.8

2

4

8

.9rt

st

2

54

85

4

3610

ba

ba

SOLUTIONS

62232223 8199 babaab

2

8224

r

ts

r

st

824 33

5

5

t

s

#7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent.

1mm

xx

So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent.

Ha Ha!

If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!

933

1

125

1

5

15

22

33

and

#8: Zero Law of Exponents: Any base powered by zero exponent equals one.

0 1x

1)5(

1

15

0

0

0

a

and

a

and

So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.

Try these:

022.1 ba

42.2 yy

15.3 a

72 4.4 ss

4323.5 yx

042.6 ts

122

.7x

2

5

9

3

3.8

2

44

22

.9ts

ts

2

54

5

4

36.10

ba

a

SOLUTIONS

022.1 ba

15.3 a

72 4.4 ss

4323.5 yx

042.6 ts

1

5

1

a54s

12

81284

813

y

xyx

1

122

.7x

2

5

9

3

3.8

2

44

22

.9ts

ts

2

54

5

4

36.10

ba

a

SOLUTIONS

4

41x

x

8

824

3

133

44222 tsts

2

101022

819

a

bba