MAP Math
Exponents &
Radicals
Study Guide
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Exponents
The exponent of a number indicates how many times the number is multiplied by itself, for example
73 is called “seven to the power of three” and equals 7 × 7 × 7.
Exponents Rules
Rule
Example
1
𝑎0 = 1 , a ≠ 0
30 = 1
2
0 𝑥 =0, x ≠ 0
03 = 0
3
1𝑥 = 1
13 = 1
4
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
75 ∙ 73 = 78
5
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
94
= 94−3 = 91
93
6
(𝑎𝑚 )𝑛 = 𝑎𝑚∙𝑛
(43 )2 = 43∙2 = 46
7
𝑎−𝑛 =
1
𝑎𝑛
8
(𝑎 ∙ 𝑏)𝑛 = 𝑎𝑛 ∙ 𝑏𝑛
9
𝑎 𝑛 𝑎𝑛
( ) = 𝑛
𝑏
𝑏
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3−2 =
1
32
(2 ∙ 5)2 = 22 ∙ 52
(
11 5 115
) = 5
7
7
2
Negative Base
When the base is a negative number, you should use parenthesis to avoid confusion, for example:
1. (−2)2 = (−2) ∙ (−2) = 4
2. (−1)3 = (−1) ∙ (−1) ∙ (−1) = −1
Tip: When the base is a negative number, the number in the exponent is enough to determine if the
result will be a positive number or a negative number:
▪
▪
If the exponent is an even number → the result will be a positive number.
If the exponent is an odd number → the result will be a negative number.
The Exponent 2
Since 2 is a common exponent, it is highly recommended to memorize its common results:
12 = 1
92 = 81
22 = 4
102 = 100
32 = 9
112 = 121
42 = 16
122 = 144
52 = 25
132 = 169
62 = 36
142 = 196
72 = 49
152 = 225
82 = 64
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Radicals
The radical sign is a symbol for the square root. Another way of understanding a square root is:
2
√𝑥 ∙ √𝑥 = (√𝑥) = 𝑥
Extracting a square root is the opposite of raising a number to the power of 2. Since the result of
raising a number to the power of 2 is always positive, you can only extract the square root of a
positive number.
Radicals Rules
Rule
Example
1
√𝑎 ∙ 𝑏 = √𝑎 ∙ √𝑏
√4 ∙ 16 = √4 ∙ √16
2
𝑎 √𝑎
√ =
𝑏 √𝑏
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√
9
√9
=
16 √16
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Solving Equations Using Radicals
When taking square roots from a number, you always have two possible solutions.
When you extract the square root of both sides of an equation, you may “lose” results.
For example:
Solve for x: 𝑥 2 = 64
Answer:
𝑥 2 = 64 // √ Extracting the square root from both sides of the equation
𝑥=8
x = 8 is a correct answer.
However, x = - 8 is also a correct answer. Let’s verify it by substituting x in the equation with – 8 :
𝑥 2 = 64 → (−8)2 = (−8) ∙ (−8) = 64
Therefore, when extracting square roots from both sides of an equation it is important to make sure
you do not “lose” any answers.
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