Download Exponent Notation If α is a real number and n is a positive integer

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Exponent Notation
If α is a real number and n is a positive integer, then αn = α ∙ α ∙ α ∙… ∙ α
Where n is the exponent and α is the base.
n times
The expression αn is read “α to the nth power.”
Rules of Exponents
Let α and b be real numbers, variables, or algebraic expressions, and let m and
n be integers. (All denominators and bases are nonzero.)
Product Rule for Exponents
𝑎𝑛 𝑎𝑚 = 𝑎𝑛+𝑚
When multiplying like bases, keep the same base and add the exponents.
Example:
Simplify using the rules of exponents
k4.k3.k6
The base is the same
k4+3+6
Keep the same base and add the exponents
k13
Example:
Simplify using the rules of exponents
(-5p4r.3)(-9p5r)
(-5)(-9)p4+5r3+1
Multiply the coefficients, keep the same base
and add the exponents for both variables
45p9r4
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Exponents
2
Quotient Rule for Exponents
𝑎𝑚
= 𝑎𝑚−𝑛
𝑛
𝑎
When dividing like bases, keep the same base and subtract the exponent of the
denominator from the exponent of the numerator.
Example:
Simplify using the rules of exponents
𝑦7
𝑦2
y7-2
Keep the same base and subtract the exponents
y5
Example:
Simplify using the rules of exponents
−4𝑥 3 𝑦 4
12𝑥 2 𝑦 2
−𝑥 3−2 𝑦 4−2
3
Simplify the coefficients and subtract the
exponents
−x𝑦 2
3
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Exponents
3
Power Rules for Exponents
(𝑎𝑛 )𝑚 = 𝑎𝑛𝑚 (1)
To raise a power to a power, multiply the exponents.
(𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑚 (2)
To raise a product to a power, raise each factor to that power
𝑎 𝑛
𝑎𝑛
(𝑏 ) = 𝑏 𝑛
(3)
To raise a quotient to a power, raise the numerator and the denominator to
that power.
Examples:
Simplify using the rules of exponents
(-6x2)3
(-6)3 (x 2)3
Quotient Rule (2)
-216x2·3
(-6)3 = -216. For x multiply the powers
-216 x6
Example:
Simplify using the rules of exponents
6
−2𝑎4
(
)
𝑏5
𝑎4.6
(-2)6(
𝑏5.6
Quotient Rule (3)
)
64a24
b30
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Exponents
4
Zero Exponent
If a is any non-zero real number then:
a0 = 1
Example:
Simplify using the rules of exponents
(-3x2)0
1
Example:
Simplify using the rules of exponents
2x(-x)0
just –x is raised to the 0 power
2x
Example:
-60
-1
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The base is 6 not -6
Exponents
5
Negative Exponent
For any positive integer n and any non-zero real number a
𝑎−𝑛 =
𝑎−𝑛
𝑏−𝑚
1
𝑎𝑛
=
𝑏𝑚
𝑎𝑛
(1)
(2)
Example:
Simplify using the rules of exponents
3-3
1
33
1
27
Example:
Simplify using the rules of exponents
-5z-3
The base is z
-5
z3
Example:
Simplify using the rules of exponents
(3a)-2
The base is 3a
1
(3a)2
1
9a2
Example:
Simplify using the rules of exponents
x2 . x-5
Use Product Rule First
x2 + (-5)
x-3
1
x3
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Exponents
6
Special Rules for Negative Exponents
For any positive integer n and any non-zero real number a
1
= 𝑎𝑛
𝑎−𝑛
Example:
Simplify using the rules of exponents
3-3
2a-2
a2
2·33
2 remains in the denominator
a2
2·27
a2
2·27
a2
54
Example:
Simplify using the rules of exponents
x
x-2
Use Quotient Rule first
x1 – (-2)
x3
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Exponents
7
Special Rules for Negative Exponents
(Intermediate Algebra)
Any non-zero number raised to the negative nth power is equal to the reciprocal
of that number raised to the nth power.
1 𝑛
𝑎−𝑛 = ( )
𝑎
𝑎 −𝑛
(𝑏 )
𝑏 𝑛
= ( )
𝑎
(1)
(2)
Example:
Simplify using the rules of exponents
5-3
Rule (1)
1 3
(5)
1
125
Example:
Simplify using the rules of exponents
1 −4
(2)
Rule (2)
24
16
Examples Using Different Rules of Exponents:
1
32 . 3-3
3
1
x-4y2
𝑦3 𝑥2
y5x-2
2
 3x 2   4 x3 

  2 
 y  y 
−4𝑎5 𝑏4
( 24𝑎𝑏−7 )
1
−2
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9𝑥
4𝑦 4
36
𝑎8 𝑏22
Exponents