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Math 142 — Rodriguez
Lehmann — 4.1
Properties of Exponents
I. Definition of an exponent
xn = x⋅
n is the exponent; x is the base
x⋅
x ⋅....x


n times
Examples:
1) 52 = 5⋅5=25
2) x3 = x⋅x⋅x
3) x2y3 = x⋅x⋅y⋅y⋅y
(52 ≠5i2)
II. Rules for Exponents
xm⋅xn =
x3⋅x2=
Product Rule
xm
Quotient Rule
xn
=
x≠0
Power Rule
(Powers to Powers)
(xm)n =
Products to Powers
(ab)m =
Quotients to Powers
⎛ a⎞
⎜⎝ b ⎟⎠
m
b≠0
Observations:
• x = x1
•
x0 = 1, x ≠ 0
ex:
(16y)0
16y0
II. Negative Exponents
A. We can deal with dividing exponents in two ways:
Use the definition of fractions:
Example:
x
3
x5
=
Use the rules of exponents:
x⋅x⋅x
=
x⋅x⋅x⋅x⋅x
x3
x5
=
Since answers must be the same we conclude:
B. We can generalize this as:
x−n =
1
x−n
=
C. Examples with negative exponents
1.
7–2
2.
1
x2
6.
x −3
2 −3
7.
–3–4
3.
x–4
8.
(–3)–4
4.
y4⋅y–6
9.
5.
(x2)–4
x–3⋅x–5
10.
x4
x −4
III. Simplifying Expressions with Exponents
To simplify an expression with exponents:
• remove parentheses
• each base appears only once
• no negative exponents
• fractions are reduced
Examples:
1.
( 4x y ) ( 2x y )
7
−5
−2
−6
(12b c )( 2b
−3 10
2.
Lehmann — 4.1
−5 −6
−60b6 c −2
⎛ −20x 3 y 3 ⎞
3. ⎜
⎟
⎝ 4 x 4 y −3 ⎠
−2
8
c
)
3
4.
5.
5x 8 y −6
( 4 x y )(10x
−3 5
⎛ −20a10 b6 ⎞
⎜
14 −2 ⎟
⎝ 10a b ⎠
⎛ 4a −5 b4 ⎞
6. ⎜
⎟
⎝ 12a11b−6 ⎠
−4
y −5
)
3
0
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IV. Scientific Notation
A. Scientific notation is used to write very large and very small numbers.
B. A number is said to be written in scientific notation if it is of the form
N x 10k , where k is an integer and 1 ≤ |N| < 10
C. Scientific notation to decimal notation
Examples:
1) 3.1 x 104 = 3.1 x 10,000 =
The decimal point got moved to the
2) 4.5 x 10 3 = 4.5 x
−
To generalize then:
If the exponent is positive, move the decimal point to the
If the exponent is negative, move the decimal point to the
3) −2.5 x 106 =
4) 5.25 x 10 5 =
−
5) 6.1 x 103 =
6) 9.4 x 10 4 =
−
D. Decimal notation to scientific notation
Steps:
(Note: my explanation is DIFFERENT than the books)
1. Determine the numerical factor, N, by looking at the decimal number and placing
the decimal point so that 1 ≤ |N| < 10.
2. Determine the exponent k by counting how many places the decimal point was
moved to get from the numerical factor N to the original decimal number.
• Make the exponent positive if you have to move the decimal point in the
numerical factor N to the right to get the decimal number.
• Make the exponent negative if you have to move the decimal point in the
numerical factor N to the left to get the decimal number.
Examples:
1) 35,000,000 =
2) 0.0000000125=
3) −0.00000000038 =
4) 0.00000087 =
5) 425,000,000 =
Lehmann — 4.1
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V. Exponential functions
A. An exponential function is a function of the form f(x) = abx or y = abx where
a≠0, b > 0 and b ≠ 1. The constant b is called the base.
f(x) = 2x
x
–3
–2
–1
0
1
2
3
f(x)
B. Function notation with exponential functions
Let f(x)=3x. Find the following:
Let h(x)=5(2)x. Find the following:
a. f(4)
a.
h(4)
b. f(–2)
b.
h(–3)
c. f(2a)
d. f(a+2)
Let g(x)=5x. Find the following:
a. g(3)
b. g(2a)
c. g(a+4)
Lehmann — 4.1
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