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