RULES FOR EXPONENTS
Exponents are used to write repeated multiplication of the same factor. In the exponential expression
23, the exponent 3 tells us how many times the base 2 is used as a factor: 23 = 2 • 2 • 2. Similarly, in
the expression x4, the exponent tells us there are four factors of the base x: x4 = x • x • x • x.
When you simplify exponential expressions, it is important to understand the difference between the
expressions −32 and (−3)2. The lack of parentheses in the expression −32 means the exponent only
raises the base “3” to the 2nd power. So −32 evaluates as −(3 • 3). On the other hand, the use of
parentheses in the expression (−3)2 means the exponent raises the base “−3” to the 2nd power. So
(−3)2 evaluates as (−3) • (−3).
1.
PRODUCT RULE: am • an = am + n
To multiply exponential expressions with the same base, keep the base, add the exponents.
Examples:
1)
23 i 24 = 2 ⋅ 2 ⋅ 2 i 2 ⋅ 2 ⋅ 2 ⋅ 2 = 27
We get the same result when we add the powers:
23 i 24 = 23 + 4 = 27
2)
p i p6 i p8 = p1 + 6 + 8 = p15 ← When an exponent is not written, the power is “1”.
3)
( − 3 ) i ( −3 )
4)
−2x4 (6x) (4x3) = (−2 • 6 • 4) (x4 + 1 + 3) = −48x8 ← Use the associative property to multiply the
2
6
= ( −3 )
2+6
= ( −3 ) = 38 ← A negative number raised to an even power is positive.
8
coefficients and add the variable powers.
2.
QUOTIENT RULE:
am
= am − n
an
To divide exponential expressions with the same base, keep the base, subtract the exponents.
a)
When the power in the numerator is greater than the power in the denominator (top heavy),
the remainder goes in the numerator.
Examples:
1)
2 5 2 i 2 i 2 i 2 i 2 23
=
=
= 23 = 8
2i2
1
22
Notice, we get the same result when we subtract the power in the denominator from the
power in the numerator:
25
= 25 − 2 = 2 3 = 8
2
2
PBCC
1
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2)
( −5 ) = −5 7 − 4 = −5 3 = −125
( )
( )
4
( −5 )
3)
x12
= x12 − 7 = x 5
x7
4)
9x 4 y 5 9 x 4 − 2 y 5 − 1 3x 2 y 4
=
=
= 3x 2 y 4
2
3x y
3
1
7
b)
← A negative number raised to an odd power is negative.
When the power in the denominator is greater than the power in the numerator (bottom
heavy), the remainder goes in the denominator and a “1” goes in the numerator when
necessary.
Examples:
1)
24
2i2i2i2
1 1
=
= 2 =
6
2
2
2
2
2
2
4
i
i
i
i
i
2
2
We get the same result when we subtract the power in the numerator from the power in
the denominator:
24
1
1 1
= 6−4 = 2 =
6
4
2
2
2
2)
( −4 ) = 1 = 1
8
8−3
5
( −4 )
( −4 ) ( −4 )
3)
k7
1
1
= 11− 7 = 4
11
k
k
k
4)
−
5)
36x 2 y 3
36 y 3 − 1
3y 2
3y 2
=
=
= 2 ← The x’s are bottom heavy; the y’s are top heavy.
12x 4 y
x
12 x 4 − 2 1 i x 2
3
PBCC
7m2n5
7
1
=−
=−
8 10
8 − 2 10 − 5
28m n
4m6n5
28 m n
2
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3.
POWER RULE: (am)n = am • n
To raise an exponential expression to a power, keep the base, multiply the exponents.
Examples:
1)
(2 )
4
3
= 24 i 24 i 24 = 24 + 4 + 4 = 212
We get the same result when we multiply the powers:
(2 )
4
4.
3
= 24 i 3 = 212
2)
( −7 ) = ( −7 )
3)
(x )
3
2
4
5
3i5
= ( −7 ) = −715
15
= x2 i 4 = x8
POWER RULES FOR PRODUCTS AND QUOTIENTS
a)
Power Rule for Products: (ab)n = anbn
To raise a product to a power, distribute the power to each base and multiply the exponents.
Examples:
1)
(x y)
2)
( −2p qr ) = ( −2)
4
3
4
= x 4 i 3 y1 i 3 = x12 y 3
2
4
1i 4
p4 i 4 q1 i 4r 2 i 4 = ( −2 ) p16 q4r 8 = 16p16 q4r 8
4
Think: “a negative raised to an even power is always positive.”
n
an
⎛a⎞
b) Power Rule for Quotients: ⎜ ⎟ = n
b
⎝b⎠
To raise a quotient to a power, distribute the power to the numerator and the denominator and
multiply the powers.
Examples:
4
1)
PBCC
⎛ 5 ⎞
5 4 625
⎜⎜
⎟⎟ =
=
x8
x8
⎝ x2 ⎠
2)
3
2
2 4 2
4 2
⎛ − 2x 2 y ⎞
⎟ = 2 x y = 4x y
⎜
⎜ z3 ⎟
z6
z6
⎠
⎝
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5.
EXPONENTS OF “1” AND “0”
a)
POWER OF “1:" a1 = a
Any number or variable raised to the 1st power is the number or variable.
Examples:
1)
b)
31 = 3
2)
x1 = x
3)
( ab )
1
= a1 b1 = ab
POWER OF “0:" a 0 = 1
Any number or variable raised to the zero power always equals 1.
When a number or variable is divided by itself, the result is 1. For example,
4
= 1;
4
−3
= 1;
−3
y
= 1;
y
−2t
= 1;
−2t
We get the same result when we apply the quotient rule:
25
= 25 − 5 = 20 = 1
5
2
Examples:
6.
1)
50 = 1
2)
x0 = 1
3)
−2y 0 = −2(1) = −2
4)
−4 0 − 5 0 = − 1 − 1 = −2
5)
j4 • j0 • j5 = j4 + 0 + 5 = j9
RULE FOR NEGATIVE EXPONENTS; a − n =
1
an
A negative exponent means “take the reciprocal of.” To simplify negative exponents, take the
reciprocal of the base (flip) and make the power positive.
For example, 2−3 is read "take the reciprocal of 23." Since 23 = 8, its reciprocal is
2−3 =
PBCC
4
1
1
, or :
3
2
8
1 1
=
23 8
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a)
If the power in the numerator is negative, move it to the denominator, and make the power
positive.
Examples:
1)
6−3 =
1
6
3)
b)
3
=
1
216
x −2
1
=
y
x2y
1
2)
y−5 =
4)
2a 3 b − 4 =
y5
2a 3 b −4 2a 3
=
1
b4
If the power in the denominator is negative, move it to the numerator, and make the power
positive.
Examples:
1
1)
5
3)
c)
−2
x5
y −3
= 5 2 = 25
1
2)
p
=
x 5 y3
= x 5 y3
1
4)
= p7
−7
5
m −3 n − 2
=
5m 3 n 2
= 5m 3 n 2
1
If the numerator and the denominator have negative factors, simplify the negative powers first.
Move negative powers in the numerator to the denominator; move negative powers in the
denominator to the numerator.
Examples:
1)
2−4 32
9
= 4 =
−2
3
2
16
3)
3−2 xy −3
xi x 4
x5
=
=
x −4 y 3
32 y 3 i y 3 9y 6
2)
x −7
y −8
=
y8
x7
Simplify negative exponents first, and then apply the product and quotient rules as
needed.
4)
⎛ 5p2 ⎞
⎜ 3 ⎟
⎝ q ⎠
−2
2
⎛ q3 ⎞
q6
q6
=⎜ 2⎟ = 2 4 =
5 p
25p4
⎝ 5p ⎠
Taking the reciprocal of the inside expression (“flipping it”) makes the outside power
positive.
5)
PBCC
−2p−4
(p )
3
−2
=
−2p−4 −2p6
= 4 = −2p2 ← Because −2 is not raised to a power, the result is negative.
p −6
p
5
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