ExponentsExponents

35ExponentialForm

base

exponent

3 3 means that is the exponential

form of t

Example:

he number

125 5 5

.125

53 means the product of 3 factors of 5 = 5 x 5 x 5

The Laws Of Exponents

The laws of exponents tell us how to simplify expressions involving exponents.

There are 7 laws of exponents.

Exponential form: raising a base to a power stands for repeated multiplication of the base times itself.

n

n times

x x x x x x x x

3Example: 5 5 5 5

n factors of x

Standard formExponential form

QuestionsWhat is a power?What is a base?What is exponential form?What is standard form?What are the laws of exponents?How many laws of exponents are there?

Write in standard form.

Write in exponential form.

Give an example of a power.Give an example of a base.Give an example of exponential form.Give an example of standard form.

Questions

The Laws of Exponents:The Laws of Exponents:

#1: Multiplying Bases: If you are multiplying bases with the same base, KEEP the BASE & ADD the EXPONENTS!

m n m nx x x

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

512

2222 93636

Try these:

22 33.1

42 55.2

25.3 aa

72 42.4 ss

32 )3()3(.5

3742.6 tsts

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

QuestionsWhat is the first law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#3: Dividing Bases: When dividing bases 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 bases, you subtract the exponents!

16

222

2 4262

6

Try these:

4

12

.7s

s

5

9

3

3.8

44

812

.9ts

ts

54

85

4

36.10

ba

ba

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

QuestionsWhat is the second law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#3: Power of a Power: If you are raising a power to a power, you multiply the exponents!

nm mnx x

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

62323 55)5(

Try these:

523.1

43.2 a

523.1

43.2 a

SOLUTIONS

10312a

QuestionsWhat is the third law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#4: Product Rule of Exponents: Raising a product to a power means raising all of the factors to the power and then multiplying.

n n nxy x y So, the power of the product is the product of the powers.

222)( baab

Try these:

322.3 a

23522.4 ba

22 )3(.5 a

342.6 ts

322.3 a

23522.4 ba

22 )3(.5 a

342.6 ts

SOLUTIONS

6323 82 aa

6106104232522 1622 bababa

4222 93 aa

1263432 tsts

QuestionsWhat is the fourth law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#5: Quotient Rule of Exponents: Raising a quotient to a power means raising the numerator and denominator to the power.

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:

5

.7t

s

2

5

9

3

3.8

2

4

8

.9rt

st

2

54

85

4

36.10

ba

ba

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

QuestionsWhat is the fifth law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#6: Law of Negative Exponents: A base raised to a negative power is the reciprocal of the base to the positive power.

1mm

xx

So, when I have a negative exponent, I take the reciprocal and change the power to a positive.

933

1

125

1

5

15

22

33

and

Try these:

42.2 yy

15.3 a

72 4.4 ss

4323.5 yx

122

.7x

2

5

9

3

3.8

2

44

22

.9ts

ts

2

54

5

4

36.10

ba

a

SOLUTIONS

15.3 a

72 4.4 ss

4323.5 yx

5

1

a54s

12

81284

813

y

xyx

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

QuestionsWhat is the sixth law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.

#7: Zero as an exponent: Anything to the zero power is 1.

0 1x

1)5(

1

15

0

0

0

a

and

a

and

So whenever I raise anything to the zero power it is always 1!

Try these:

022.1 ba

042.6 ts

SOLUTIONS

022.1 ba

042.6 ts

1

1

QuestionsWhat is the seventh law of exponents?When can I use it?When can I not use it?Give an example of where I use it.Give an example of where I can’t use it.Why does it work? Give an example.