Properties of Exponents - WordPress.com · integer exponents. 4. 4x2y 2xy2 22x y xy2 2xy 2y 2x y Simplify 4 divided by 2. The “x”s have the same base, subtract their exponents:

Part 1: Multiplication Properties

Part 2: Division Properties

Part 3: Integers & Exponents

Part 4: Simplify Expressions with Exponents

Properties of Exponents

Part 1

Multiplication Properties

Exponent Review

An exponent is a number placed above and to the right of

another number to show that it has been raised to a power.

This is 2 to the fifth power. Which means:

2 • 2 • 2 • 2 • 2

25 = 32

25 Base

Exponent

Multiplying Rule #1

35 = 3 • 3 • 3 • 3 • 3

38 = 3 • 3 • 3 • 3 • 3 • 3 • 3 • 3

How many 3s all together? 13

Words Numbers Algebra

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

MULTIPLYING POWERS WITH THE SAME BASE

35 • 38 = 35 + 8 = 313

bm • bn = bm + n

Practice: Multiply. Write the product as one power.

1. 66 • 63

66+3

69

2. n5 • n7

n5+7

n12

3. 25 • 2

25+1

26

4. 244 • 244

244+4

248

5. 810 • 83 • 87

810+3+7

820

6. h4 • m4

Don’t have the

same base.

h4 m4

Multiplying Rule #2

Power of powers with the same base:

For example: (43)5

43 = 4 • 4 • 4

Now do that 5 times.

(4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4) (4 • 4 • 4)

How many 4s do you have?

15

(43)5 = 415

The rule is that if you are taking a power to another power,

you multiply the exponents.

Multiplying Rule #3

The Power of a Product Property is used when you have

different bases being raised to another power.

In this case, distribute the POWER to each base by

multiplication.

For example:

(23m5)4

You are going to take (23)4 and (m5)4

(23m5)4 = 212m20

Practice:

1. (32y3)2

3(2•2)y(3•2)

34y6

81y6

2. (5a2b4)5

5(1•5)a(2•5) b(4•5)

55a10 b20

55 is way more than 1,000

so leave it as 55.

3. (-4xy3)3

(-4)(1•3)x(1•3) y(3•3)

(-4)3x3y9

-64x3y9

4. (24h3k2)2

2(4•2)h(3•2) k(2•2)

28h6 k4

256h6 k4

Part 2

Division Properties

Dividing Rule #1

69 = 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6

64 = 6 • 6 • 6 • 6

How many 6s are left?

This is why the answer is 65

DIVIDING POWERS WITH THE SAME BASE


To divide powers with the same base, keep the base and subtract the exponents.

69

64 = 69-4 = 65 = bm-n bm

bn

5

Practice: Divide. Write the quotient as one power.

1. 75

73

75-3

72

2. x10

x9

x10-9

x

3. 99

92

99-2

97

4. 43

43

43-3

40 = 1

4 • 4 • 4

4 • 4 • 4

= 1

1

Any number to the

zero power is 1.

5. e10

e5

e10-5

e5

6. 7017

705

7017-5

7012

Division Rule #2

The Power of Quotient Property has you distribute the

power to every number inside the parentheses – both the

numerator and the denominator get the power.

For example

3

2 2

= 22

32

= 4

9

Practice:

6

x 3

= x3

63

= x3

216

4

2m 4

= 24m4

44

= 16m4

256

1.

2.

Reduce the

numbers of

the fraction

part of the

answer: 16/256

16/256 reduces

to 1/16.

= m4

16

Practice: 3.

4.

= (-2)3

33

= -8

27

2

8b 3

= 83b3

23

= 512b3

8

= 64b3

Part 3

Integers & Exponents

NEGATIVE EXPONENTS


A power with a negative exponent equals 1 divided by that power with it’s opposite exponent.

b–n = 1 bn 5–3 = =

1

125

1 53

You can NEVER leave an answer with a negative exponent.

You must have all positive exponents in your final answer.

Practice simplifying products and quotients

that may have negative exponents.

1. 2–5 • 23

2 –5+3

2 –2

22

1

1 4

Bases are the same, so add the exponents.

6 5–8

6 –3

1 63

65 68

216

1

2. Bases are the same, so subtract the exponents.

Practice simplifying products and quotients

that may have negative exponents.

5 2–3

5 –1

1

5 1

52 53

1 5

3. Bases are the same, so subtract the exponents.

7 –6+7

7 1

7

4. 7–6 • 77 Bases are the same, so add the exponents.

Part 4

Simplify Expressions with Exponents

Simplifying Expressions with Exponents means…

You need to know ALL of the properties of exponents

and

you need to know when to use them.

For example: Simplify the following expression.

4c0

4(1)

4

This is 4 times c to the zero power.

Order of operations states that you

have to solve exponents before

multiplying.

Anything to the zero power is 1.

Practice simplifying expressions with

integer exponents.

1. (2m3 • 6m4)2

(12m7)2

122m14

144m14

Simplify what is in parenthesis.

2 • 6 = 12

m3 • m4 = m7

Distribute the Power of 2 to

everything in the parenthesis.

Solve 122.


integer exponents.

2. 3y2(2y)3

= 3y2 23y3

= 3 23y5

= 3 8y5

= 24y5

Distribute the Power of 3 to


The “y”s have the same base, add

the exponents.

Solve 23.

Multiply 3 and 8.


integer exponents.

3. (6j2k4)-3

6-3j-6k-12

1 . 63 j6k12

1 .

216j6k12

Distribute the Power of -3 to


You cannot have negative

exponents in the answer. To

make them positive you need to

move them to the denominator

(take the reciprocal).

Solve 63.


integer exponents.

4. 4x2y

2xy2

2x2y

xy2

2xy

y2

2x

y

Simplify 4 divided by 2.

The “x”s have the same

base, subtract their

exponents: 2 – 1 = 1. Since

this is positive, x1 stays in

the numerator.

The “y”s have the same

base, subtract their

exponents: 1 – 2 = -1.

Since this is negative, y-1 it

moves to the denominator.


integer exponents.

4 Before distributing the Power of 4 to

everything in the parentheses, first

simplify the variables.

The “x”s have the same base, subtract

their exponents. 7 – 2 = 5. Since it is

positive 5, x5 stays in the numerator. 4

The “y”s have the same base, subtract

their exponents. 3 – 1 = 2. Since it is

positive 2, y2 stays in the numerator.

4 Distribute the Power of 4 to everything

in parentheses.


integer exponents.

-2 Before distributing the Power of -2 to

everything in parentheses, first reduce what is

in parentheses.

Reduce 27/18.

They are both divisible by 9.

Which reduces to 3/2.

The “w”s are the same base. Since they are

being divided, subtract their exponents.

-4 – 6 = - 10

Since it is a negative exponent, “w” is in the

denominator with a positive 10 exponent.

-2

-2

The “x”s are the same base. Since they are

being divided, subtract their exponents.

3 – -5 = 8

Since it is a positive exponent, “x” is in the

numerator with a positive 8 exponent.

-2

6.

#6 Continued…

-2 Now distribute the Power of -2 to everything

in parentheses.

You cannot have negative exponents in the

answer. To make them positive you need to

take the reciprocal.

Move the terms with negative exponents from

the numerator to the denominator.

Move the terms with negative exponents from

the denominator to the numerator.

Solve 22 and 32.

The key to these exponent problems is to

take your time.

Keep all of the rules handy so they are easy

to reference.

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Properties of Exponents - WordPress.com · integer exponents. 4. 4x2y 2xy2 22x y xy2 2xy 2y 2x y Simplify 4 divided by 2. The “x”s have the same base, subtract their exponents: