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```Apply Properties
of Rational
Exponents
Section 8.2
MATH 116-460
Mr. Keltner
Properties of
Rational Exponents
п‚§ 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
п‚§ A rational exponent is an exponent
that is a fraction.
п‚§ When the numerator is 1, then we can
generalize as such:
1
a n пЂЅ n a
п‚§ Where n is a natural number (one that can
be counted on fingers) other than 1.
п‚§ 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
п‚§ On the previous slide, the last example
showed a variable power raised to
another power.
п‚§ Using properties of exponents, we
were able to simplify to a single term
or expression.
п‚§ In general, rational exponents can be
simplified by the following property:
п‚§ If aп‚і0 and m and n are natural numbers
other than 1:
m
a
n
пЂЅ
n
a
m
пЂЅ
пЂЁ aпЂ©
n
m
Example 2
п‚§ 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
п‚§ We are still allowed to apply the negative
exponent property to using radicals.
п‚§ This property states that:
1
пЂ­m
n
a
пЂЅ m
a n
п‚§ 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
п‚§ Keep in mind the вЂњRoots and
BranchesвЂќ analogy used earlier.
п‚§ We have also seen properties of
exponents that could allow us to
convert from radical expressions
to exponential expressions.
п‚§ Example 4: Simplify.
5
y
1
2
3
4
x
3
пЂЁ3 y пЂ­ 2 пЂ©
5
Example 5
п‚§ Simplify each expression, using
rational exponents.
п‚ пЂ 8
п‚ пЂ 4
36
8
x
6
6
x y
2
Assessment
Pgs. 555-557:
#вЂ™s 9 - 99, multiples of 9
```
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