Reviewing the Exponent Laws

• Throughout the study of all modern sciences, extremely large and extremely small numbers frequently appear.

The distance from the Earth to the Sun

150 000 000 000 m

1 000 000 000 000 000

Number of cells in the human body

An iPad can hold up to

32 000 000 000 bytes of information

In an attempt to be more efficient when operating on these numbers, a series of mathematical shortcuts

were derived.

These shortcuts evolved into the Exponent Laws

• The Exponent Laws are one set of tools a mathematician can use to help make work

quicker and easier.

1. Multipliying PowersSimplify the following powers

(click to see the answer)

1. 23 = 2 X 2 X 2= 8

2. 52 = 5 X 5= 25

NOTE:

25

baseexponent

Now we try this!!!

6. 24 X 23 We expand each expression

= 2 X 2 X 2 X 2 X 2 X 2 X 2

How many 2s are you multiplying together? 7

= 27

Notice the exponents

3. 24 X 23

Is there a way would could get 7 given the initial exponents of 4 and 3

= 27

Correct! You can add the exponents together

This property holds true for multiplying powers with the same base.

Express as a single power.(click to see each answer)

3. 25 X 22 =

4. 32 X 34 =

5. 17 X 13 =

27

36

110

ExLaw #1

Let B be any baseLet x and y be any exponent

(Bx)(By) = Bx + y

“When multiplying powers with the same base, add the exponents!!!”

For example:

(x3) (x8) = x11

(a4) (a3) = a7

Consider the Division of powers

25

23Expand each power=

2 X 2 X 2 X 2 X 2

2 X 2 X 2

Notice: There are now numbers on the top and the bottom that can be divided out!!!

1

1

1

1

1

1 = 2 X 2

= 22

Notice the exponents:

25

23=

Is there anything you could do with a 5 and a 3 to get 2?

22

Subtract! That is correct!This is true for any division of powers with the same base

Reduce the following to a single power

27

23=

1. 27 - 3 = 24

43

41=

2. 43 - 1 = 42

56

52=

3. 56 - 2 = 54

ExLaw #2Let B be any baseLet x and y be any exponent

(Bx) = Bx - y

(By)

“When dividing powers with the same base, subtract the exponents!!!”

3. Power of PowersExpand the following:

Sometimes the base you are expanding is a power itself!

23 = 2 X 2 X 2

Expand the following:

Expand this in the same way

(22)3 22 X 22 X 22=

= 2 X 2 X 2 X 2 X 2 X 2 Which can be written as …

= 26

How many 2s are you multiplying?... 6

Examine the exponents

(22)3 = 26

What can you do with 2 and 3 to get 6?

Multiply! Correct!This property is true for all power of powers with the same base.

Reduce the following to a single power

(27)2=

1. 27 X 2 = 214

ExLaw #3

Let B be any baseLet x and y be any exponent

(Bx)y = B(x X y)

“When expanding a power of powers, multiply the exponents!!!”

To understand the fourth and fifth exponent laws,

examine the following pattern

ExLaw #4x0 = 1

23 = 8

22 = 4

21 = 2 Continue the pattern

20 =

1 2

x

1 2

x

1 2

x 1

“Any base to the exponent zero equals 1”

Is there a pattern?

2-1 = 121

2-2 =12

12

X =1

22

2-3 =12

12

12

X X =1

23

Keep going….

1

4

1

8

=

=

ExLaw #5

X-n = 1Xn

“Eliminate any negative exponents by inverting the

power”

For Example: Simplify

(7)0 = 1

3-2 =1

32=

1

9

8-1 = 1

81

1

8=

1 =

8-2

82

1= 64

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