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Rational Exponents

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How Do We Use Rational
Exponents?
• Do Now: Perform the indicated operation
and simplify
1.
2.
1
nth Roots
nth Roots
An nth root of number a is a number whose nth
power is a.
n
a пЂЅ a number whose nth power is a
If the index n is even, then the radicand a must
be nonnegative.
4
16 пЂЅ 2, b u t 4 пЂ­ 16 is not a real number
5
пЂ­ 32 пЂЅ пЂ­ 2
2
age 393
Square Root of x2
x
2
пЂЅ x
3
Radicals
4
Rational Exponents
5
Exponent 1/n When n Is Even
6
When n Is Even
1
100
пЂЅ
2
100 пЂЅ 10
1
625
пЂЅ
4
4
625 пЂЅ 5
1
64
6
пЂЅ
1
пЂЁпЂ­ 4 пЂ© 2
6
пЂЅ
64 пЂЅ 2
пЂ­ 4 is not yet defined
7
Exponent 1/n When n Is Odd
8
Exponent 1/n When n Is Odd
1
пЂЅ
3
пЂЁ пЂ­ 27 пЂ©
1
27
3
3
27 пЂЅ 3
пЂЅ
3
пЂ­ 27 пЂЅ пЂ­ 3
1
пѓ¦ 1 пѓ¶5
пѓ§
пѓ· пЂЅ
пѓЁ 32 пѓё
5
1
32
пЂЅ
1
2
9
nth Root of Zero
0 пЂЅ0
n
10
Rational Exponents
11
Evaluating in Either Order
2
пЂЁ8 пЂ© 3
пЂЅ
пЂЁ 8пЂ©
2
3
пЂЅ пЂЁ2 пЂ© пЂЅ 4
2
or
2
пЂЁ8 пЂ© 3
пЂЅ
3
8 пЂЅ
2
3
64 пЂЅ 4
12
Negative Rational Exponents
13
Evaluating a-m/n
пЂЁ8 пЂ©
пЂ­
2
3
пЂЅ
1
2
пЂЁ8 пЂ© 3
пЂЅ
1
пЂЁ 8пЂ©
3
2
пЂЅ
1
пЂЁ2 пЂ©
2
пЂЅ
1
4
14
Rules for Rational Exponents
15
7
Simplifying
пЂЁy пЂ©
6
1
6
пЂЅ
пЂ­
пѓ¦
пѓ§ a 2b 3
пѓ§
пѓЁ
1
1
6
y
6
пЂЅ y
пѓ¶
пѓ· пЂЁ ab пЂ© пЂЅ
пѓ·
пѓё
16
Simplifying
пЂЁy пЂ©
6
1
6
пЂЅ
пЂ­
пѓ¦
пѓ§ a 2b 3
пѓ§
пѓЁ
1
1
6
y
6
пЂЅ y
пѓ¶
пѓ· пЂЁ ab пЂ© пЂЅ
пѓ·
пѓё
пЂ­
пѓ¦
пѓ§ a 2b 3
пѓ§
пѓЁ
1
пЂЅa
1
1
пЂ«1
1
2
b
3
2
пЂЅa b
2
3
пѓ¶ 1 1
пѓ· a b
пѓ·
пѓё
пЂЁ
пЂ©
пЂ­ пЂ«1
3
17
Simplifying
пЂЁy пЂ©
6
1
6
пЂЅ
пЂ­
пѓ¦
пѓ§ a 2b 3
пѓ§
пѓЁ
1
1
пЂЁ9 x
8
y
6
y
пЂЅ y
6
пѓ¶
пѓ· пЂЁ ab пЂ© пЂЅ a 2 b 3
пѓ·
пѓё
пЂ­ 10
3
z
12
пЂ©
2
1
2
пЂЅ
18
Multiplying Radicals – Different Indices
1
4
2пѓ—
3
2пѓ— 3 пЂЅ
2 пЂЅ 2 пѓ—2
4
1
2
1
пЂЅ2
4
пЂ«
1
2
3
пЂЅ2
4
пЂЅ
4
2 пЂЅ
3
4
8
19
Multiplying Radicals
Different Indices
1
4
2пѓ—
2 пЂЅ 2 пѓ—2
4
1
3
1
2
1
пЂЅ2
4
пЂ«
1
2
3
пЂЅ2
4
пЂЅ
4
2 пЂЅ
3
1
2 пѓ— 3 пЂЅ 23 пѓ—32 пЂЅ
20
4
8
Different Indices
1
4
2пѓ—
2 пЂЅ 2 пѓ—2
4
1
3
1
2
1
1
пЂЅ2
4
2
пЂ«
1
2
3
пЂЅ2
4
пЂЅ
4
2 пЂЅ
3
4
3
2 пѓ— 3 пЂЅ 23 пѓ—32 пЂЅ 26 пѓ—36 пЂЅ
21
8
Different Indices
1
4
2пѓ—
2 пЂЅ 2 пѓ—2
4
1
3
1
2
1
1
пЂЅ2
4
2
пЂ«
1
2
3
пЂЅ2
2
6
пЂЅ
6
2 пѓ— 3 пЂЅ
4
2 пЂЅ
3
4
3
2 пѓ— 3 пЂЅ 2 пѓ—3 пЂЅ 2 пѓ—3 пЂЅ
3
4
6
2
6
3
22
8
Different Indices
1
4
2пѓ—
2 пЂЅ 2 пѓ—2
4
1
3
1
2
1
1
пЂЅ2
4
2
пЂ«
1
2
3
пЂЅ2
2
6
пЂЅ
6
2 пѓ— 3 пЂЅ
4
2 пЂЅ
3
4
8
3
2 пѓ— 3 пЂЅ 2 пѓ—3 пЂЅ 2 пѓ—3 пЂЅ
3
4
6
2
6
3
6
108
23
Rational Exponents
Eliminate the root, then the power
2
a3 пЂЅ 2
24
Eliminate the Root, Then the Power
2
a3 пЂЅ 2
пѓ¦
пѓ§a3
пѓ§
пѓЁ
2
3
пѓ¶
пѓ· пЂЅ 23
пѓ·
пѓё
a пЂЅ8
2
a
2
пЂЅп‚± 8
a пЂЅ п‚±2 2
CHECK
25
Negative Exponents
пЂЁr пЂ­ 1пЂ©
пЂ­
2
3
пЂЅ1
26
Negative Exponents
Eliminate the root, then the power
пЂЁr пЂ­ 1пЂ©
пЂ­
2
пЂЅ1
3
пЂ­ пѓ¶
пѓ¦
пѓ§ пЂЁr пЂ­ 1пЂ© 3 пѓ·
пѓЁ
пѓё
2
пЂЁr пЂ­ 1пЂ©
2
пЂ­3
пЂЅ1
пЂ­3
пЂЅ1
пЂЁr пЂ­ 1пЂ©
2
пЂЅп‚± 1
r пЂ­ 1 пЂЅ п‚±1
r пЂЅ2
r пЂЅ0
27
CHECK
Negative Exponents
Eliminate the root, then the power
пЂЁ2 t пЂ­ 3 пЂ©
пЂ­
2
3
пЂЅ пЂ­1
28
No Solution
Eliminate the root, then the power
пЂЁ2 t пЂ­ 3 пЂ©
пЂ­
2
пЂЅ пЂ­1
3
пЂ­ пѓ¶
пѓ¦
пѓ§ пЂЁ2 t пЂ­ 3 пЂ© 3 пѓ·
пѓЁ
пѓё
2
пЂЁ2 t пЂ­ 3 пЂ©
2
пЂ­3
пЂЅ пЂЁпЂ­ 1пЂ©
пЂ­3
пЂЅ пЂ­1
пЂЁ2 t пЂ­ 3 пЂ©
2
пЂЅ п‚± пЂ­1
29
No Solution
Eliminate the root, then the power
пЂЁ2 t пЂ­ 3 пЂ©
пЂ­
2
пЂЅ пЂ­1
3
пЂ­ пѓ¶
пѓ¦
пѓ§ пЂЁ2 t пЂ­ 3 пЂ© 3 пѓ·
пѓЁ
пѓё
2
пЂЁ2 t пЂ­ 3 пЂ©
2
пЂ­3
пЂЅ пЂЁпЂ­ 1пЂ©
пЂ­3
пЂЅ пЂ­1
пЂЁ2 t пЂ­ 3 пЂ©
2
пЂЅ п‚± пЂ­1
No real solution
30
Strategy for Solving Equations with
Exponents and Radicals
31
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