School

Ivy Tech Community College, Indianapolis **We aren't endorsed by this school

Course

MATHEMATIC 023

Subject

Mathematics

Date

Nov 13, 2023

Type

Other

Pages

10

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Page 1 of 10
10.1 -Exponents
Exponent Review
Exponents mean repeated multiplication
3
5
= 3
∙
3
∙
3
∙
3
∙
3
Any base with an exponent of 0 equals 1
𝑚𝑚
0
= 1
7
0
= 1
Exponents & negative bases
−
2
4
=
−
2
∙
2
∙
2
∙
2 =
−
16
(
−
2)
4
= (
−
2)(
−
2)(
−
2)(
−
2) = 16
When no exponent is shown, the exponent is 1
𝑥𝑥
=
𝑥𝑥
1
3
4
= 81
(
−
7)
2
= 49
−
7
2
=
−
49
𝑥𝑥
0
+ 6
0
= 2
Product Rule
►
When you multiply 2 monomials, multiply the coefficients and add the exponents
2
𝑥𝑥
5
∙
7
𝑥𝑥
3
= 14x
8
−
4
𝑛𝑛 ∙
3
𝑛𝑛
4
=
−
12n
5
(4x
3
)(5x
5
)(2x
2
)
= 40x
10
► If
the monomial includes more than 1 variable, work with each variable separately
(
−
3x
3
y
2
)(
−
5x
4
y
6
)
= 15x
7
y
8
Power of a Product Rule
►
When the base is already an exponential expression, you raise the coefficient to the given
power and multiply the exponents.
Work with each variable separately.
(2
𝑥𝑥
7
)
3
= (2x
7
)(2x
7
)(2x
7
) = 8x
21
(5
𝑥𝑥
3
𝑦𝑦
2
𝑧𝑧
)
3
= 125x
9
y
6
z
3
(
−
7
𝑎𝑎
3
𝑏𝑏
3
)
2
= 49a
6
b
6

Page 2 of 10
10.1 -Exponents
Power of a Quotient Rule
►
Same as power of a product
�
2𝑥𝑥
2
𝑦𝑦
�
3
=
�
2x
2
y
� �
2x
2
y
� �
2x
2
y
�
=
8x
6
y
3
�
𝑎𝑎
2
𝑏𝑏
𝑐𝑐
3
�
5
=
a
10
b
5
c
15
�
−2𝑥𝑥
3
𝑦𝑦
7
𝑧𝑧
�
4
=
16x
12
y
28
z
4
Quotient Rule
►
Divide the coefficients. Keep the base and subtract the exponents.
12𝑦𝑦
6
2𝑦𝑦
3
=
12∙y∙y∙y∙y∙y∙y
2∙y∙y∙y
= 6
∙
y
∙
y
∙
y = 6y
3
20𝑥𝑥
7
5𝑥𝑥
2
= 4x
7−2
= 4x
5
𝑚𝑚
3
𝑚𝑚
3
= m
3−3
= m
0
= 1
𝑥𝑥
11
𝑥𝑥
3
= x
8
6
11
6
3
= 6
8
𝑥𝑥
12
𝑦𝑦
5
𝑥𝑥
8
𝑦𝑦
4
= x
4
y
8𝑎𝑎
3
𝑏𝑏
8
𝑐𝑐
3
18𝑎𝑎𝑐𝑐
3
=
4a
2
b
8
9
Practice Set 10.1
Exponents Summary Handout

Page 3 of 10
10.2 - Negative Exponents & Scientific Notation
Negative Exponents
𝑥𝑥
2
𝑥𝑥
5
=
x∙x
x∙x∙x∙x∙x
=
1
x
3
or
x
2−5
= x
−3
To simplify a base with a negative exponent, take the reciprocal of the base and
make the exponent positive
Note:
You are only taking the reciprocal of the base, not the coefficient
8
𝑚𝑚
−5
=
8
m
5
3
−2
=
1
3
2
7
𝑥𝑥
−4
= 7x
4
𝑥𝑥
−4
𝑦𝑦
−3
=
y
3
x
4
𝑥𝑥
𝑥𝑥
−7
= x
∙
x
7
= x
8
𝑥𝑥
𝑥𝑥
−7
= x
1−
(
−7
)
= x
8
You might say that if a base has a negative exponent, wherever it is, it wants to
be someplace else.
If the base is on top of a fraction, it wants to be on the
bottom.
If a base is on the bottom of a fraction, it wants to be on the top.
Review
𝑥𝑥
5
∙ 𝑥𝑥
3
=
𝒙𝒙
𝟖𝟖
When multiplying add exponents
(
𝑥𝑥
5
)
3
=
𝒙𝒙
𝟏𝟏𝟏𝟏
Exponents raised to another power are multiplied
𝑥𝑥
5
𝑥𝑥
3
=
𝒙𝒙
𝟐𝟐
When dividing, subtract exponents
5
𝑥𝑥
−3
=
𝟏𝟏
𝒙𝒙
𝟑𝟑
Take the reciprocal of bases with negative exponents
𝑥𝑥
0
=
𝟏𝟏
Anything with an exponent of 0 is 1