10 Math
Name:_______________________________
Date:________________________________
Investigating Exponent Laws
Terminology
base
7
3
exponent
power
Complete the following table, then answer the questions below.
Law #1
Exponential Form
Expanded Form
Power Form
Exponents
62 x 63
(6x6)x(6x6x6)
65
2+3=5
(-5)3 x (-5)4
(-5x-5x-5)x(-5x-5x-5x-5)
(-5)7
3+4=7
a x a3
(a)x(axaxa)
a4
1+3=4
3x3x3x3x3x3x3
3x3x3x3
33
7-4=3
-4x-4x-4
-4x-4
(-4)1
3-2=1
bxbxbxbxb
bxbxb
b2
5-3=2
(72)3
(7x7)x(7x7)x(7x7)
76
2x3=6
(23)3
(2x2x2)x(2x2x2)x(2x2x2)
29
3x3=9
(c4)2
(cxcxcxc)x(cxcxcxc)
c8
4x2=8
7
7
4
3 3 =
3
4
3
3
Law #2
3
2
(-4)  (-4) =
5
b
3
b
Law #3
(-4)
2
(-4)
What do you notice about the first three rows of the table (Law #1) ?
To multiply powers of the same base, add the exponents.
bx x by = bx + y
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What do you notice about the second three rows of the table (Law #2) ?
To divide powers of the same base, subtract the exponents.
bx  by = bx - y
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What do you notice about the third three rows of the table (Law #3) ?
To find the power of a power, multiply the exponents.
(bx)y = bxy
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Is it possible for a power to have a negative exponent?
Yes, because of the second law. For example:
b2  b5 = b2-5 = b-3
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10 Math
Name:_______________________________
Date:________________________________
Investigating Exponent Laws 2
Complete the following tables – use fractions where applicable (not decimals).
Notice that each row is one multiple of the base times the row below it. Therefore, to move down the
table, it is necessary to divide by the given base.
Base 5
Base 3
Base 2
54
5x5x5x5
625
34
3x3x3x3
81
24
2x2x2x2
16
53
5x5x5
125
33
3x3x3
27
23
2x2x2
8
52
5x5
25
32
3x3
9
22
2x2
4
51
5
5
31
3
3
21
2
2
50
1
1
30
1
1
20
1
1
5-1
1
5
1
5
3-1
1
3
1
3
2-1
1
2
1
2
5-2
1
5x5
1
25
3-2
1
3x3
1
9
2-2
1
2x2
1
4
5-3
1
5x5x5
1
125
3-3
1
3x3x3
1
27
2-3
1
2x2x2
1
8
5-4
1
5x5x5x5
1
625
3-4
1
3x3x3x3
1
81
2-4
1
2x2x2x2
1
16
What do you notice about 51, 31, and 21 ?
A base with exponent one is equal to itself (the base).
b1 = b
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What do you notice about 50, 30, and 20 ?
A base with exponent zero is equal to 1.
b0 = 1
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What do you notice about 5-1, 3-1, and 2-1 ?
A base with exponent negative one is equal to one over the base.
b-1 =
1
b
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What do you notice about 54 vs. 5-4, 32 vs. 3-2, and 23 vs. 2-3 ?
Positive and negative bases are reciprocals.
b-x =
1
x
b
and bx =
1
-x
b
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