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Apply Properties
of Rational
Exponents
Section 8.2
MATH 116-460
Mr. Keltner
Properties of
Rational Exponents
 Let a and b be real numbers and let m
and n be integers.
Property Name
Definition
Product of Powers
am п‚џ an = am+n
Power of a Power
(am)n = amп‚џn
Power of a Product
Negative Exponent
Quotient of Powers
п‚ пЂ (ab)m =
a
пЂ­m
a
m
a
n
пЂЅ
п‚ пЂ 1
3
п‚·3
2
2
ambm
п‚ пЂ 3
пЂЁ25 п‚· 16 пЂ©
пЂЅ a
5
2
п‚ пЂ m
2
2
пЂЅ2
3
2
2
1
пЂЅ
2
пЂЅ 3 пЂЅ 9
2
пЂЅ 2 пЂЅ 32
5
п‚· 16
1
1
пЂЁ5 2 пЂ­ 3 2 пЂ©
1
2
пЂЅ
2
пЂЅ 5 п‚· 4 пЂЅ 20
1
6
пЂЅ2 пЂЅ2
1
2
пѓ¦пЂ 343 пѓ¶пЂ пѓЁпЂ 729 пѓёпЂ п‚ пЂ пЂЁ5 3 п‚· 3 пЂ©
пЂЅ 25
пЂ­1
пЂЁ1 2 пЂ« 3 2 пЂ©
36
m пЂ­n
a
пѓ¦пЂ a пѓ¶пЂ пЂЅ m
пѓЁпЂ b пѓёпЂ b
1
пЂЅ2
36
п‚ пЂ m
пЂЅ 3
2
3
1
a
3
пЂЁ пЂ©
5
п‚ пЂ m
Power of a Quotient
Example
1
1
3
пЂЅ
343
3
пЂЅ
1
729
3
7
9
Rational Exponents
 A rational exponent is an exponent
that is a fraction.
 When the numerator is 1, then we can
generalize as such:
1
a n пЂЅ n a
 Where n is a natural number (one that can
be counted on fingers) other than 1.
 Example 1: Rewrite using radicals,
п‚ пЂ then simplify.
1
1
49 пѓ¶пЂ 2
1
1
пѓ¦пЂ 49 2
625 4
18 n 5
пѓЁпЂ x 10 пѓёпЂ Rational Exponents
OTHER than 1
 On the previous slide, the last example
showed a variable power raised to
another power.
 Using properties of exponents, we
were able to simplify to a single term
or expression.
 In general, rational exponents can be
simplified by the following property:
 If a0 and m and n are natural numbers
other than 1:
m
a
n
пЂЅ
n
a
m
пЂЅ
пЂЁ aпЂ©
n
m
Example 2
 Evaluate each expression:
3
32
пЂ­ 49
5
п‚ пЂ 3
2
п‚ пЂ пЂЁ3 a пЂ« 4 пЂ©
4
5
пѓ¦пЂ 1 пѓ¶пЂ пѓЁпЂ 16 пѓёпЂ 3
2
Negative Rational
Exponents
 We are still allowed to apply the negative
exponent property to using radicals.
 This property states that:
1
пЂ­m
n
a
пЂЅ m
a n
 Example 3: Simplify by rewriting as
radicals, if possible.
п‚ пЂ 49
пЂ­1
2
пЂ­ 27
пЂ­2
3
пѓ¦пЂ 16 пѓ¶пЂ пѓЁпЂ 81 пѓёпЂ пЂ­3
4
Rewriting as Rational
Exponents
 Keep in mind the “Roots and
Branches” analogy used earlier.
 We have also seen properties of
exponents that could allow us to
convert from radical expressions
to exponential expressions.
 Example 4: Simplify.
5
y
1
2
3
4
x
3
пЂЁ3 y пЂ­ 2 пЂ©
5
Example 5
 Simplify each expression, using
rational exponents.
п‚ пЂ 8
п‚ пЂ 4
36
8
x
6
6
x y
2
Assessment
Pgs. 555-557:
#’s 9 - 99, multiples of 9
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