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# Rational Exponents

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10.2
Rational Exponents
1. Evaluate rational exponents.
2. Write radicals as expressions raised to rational
exponents.
3. Simplify expressions with rational number exponents
using the rules of exponents.
4. Use rational exponents to simplify radical expressions.
Integer Exponents
5
5
5
2
0
пЂ­2
пЂЅ 25
пЂЅ 1
пЂЅ
1
5
2
пЂЅ
1
25
Integer Exponents
x
2
пѓ— x
3
2 3
пЂЁa
2
b
3 4
пЂЅ x
5
пЂЅ x
6
8
пЂЅ a b
12
Rational Exponents
16 пЂЅ
25 пЂЅ
3
8 пЂЅ
16
25
1/ 2
1/ 2
8
1/ 3
пЂЅ пЂЁ4
пЂЅ пЂЁ5
2 1/ 2
2 1/ 2
пЂЅ пЂЁ2
3 1/ 3
пЂЅ 4
пЂЅ 5
2пѓ—
2пѓ—
пЂЅ 2
1
2
пЂЅ 4 пЂЅ 4
1
1
2
3пѓ—
пЂЅ 5 пЂЅ 5
1
1
3
пЂЅ 2 пЂЅ 2
1
New notation for a familiar concept!
Denominator is the index (root) of the radical.
Evaluate:
27
1/ 3
пЂЅ
25
1/ 2
пЂЅ
32
1/ 5
пЂЅ
343
1/ 3
пЂЅ
3
27 пЂЅ 3
25 пЂЅ 5
5
3
32 пЂЅ 2
343 пЂЅ 7
Evaluate:
пЂЁпЂ­
1/ 4
пЂ­ 81
1000
пЂ­ 8
8
пЂЅ
4
пЂ­ 81
Not a real number
1/ 4
пЂЅ пЂ­ 81 пЂЅ пЂ­ 3
1/ 3
пЂЅ
1/ 3
пЂ­1 / 3
4
3
1000 пЂЅ 10
пЂЅ пЂ­ 8 пЂЅ пЂ­2
3
пЂЅ
1
8
1/ 3
пЂЅ
1
3
8
пЂЅ
1
2
22 a
w
пЂЁx
1/ 2
1/ 8
пЂЅ 22
пЂЅ
1/ 2
пЂЅ
8
a
w
x пЂ« 3
Rational Exponents
4
пЂЅ пЂЁ4
3/2
1/ 2 3
3
пЂЅ 8
Numerator is the power.
4
пЂЅ пЂЁ4
3/2
3 1/ 2
1/ 2
пЂЅ 8
Generally easier to find the root first and then the power.
x
a
power
b
root
Evaluate:
27
2/3
25
3/2
пЂ­ 81
27
5/4
пЂ­2 / 3
пЂЅ
пЂЁ
пЂЅ
пЂЁ
3
27
пЂЅ 9
25
пЂЅ 125
2
3
пЂЅ пЂ­ пЂЁ 81 пЂ© пЂЅ пЂ­ 243
5
4
пЂЅ
1
27
2/3
пЂЅ
1
пЂЁ
3
2
пЂЅ
1
9
Evaluate:
пЂЅ
пЂЁ
3
16
3/4
пЂЅ
пЂЁ
4
32
3 /5
пЂЅ
пЂЁ
5
1000
2/3
3/5
1
пЂЅ 1
1000
16
3
2
пЂЅ 100
пЂЅ 8
3
пЂ­2 / 3
a) 4
1
b) 4
c) пЂ­4
d) пЂ­
1
4
Slide 10- 11
пЂ­2 / 3
a) 4
1
b) 4
c) пЂ­4
d) пЂ­
1
4
Slide 10- 12
Write in Exponential Form:
8
33
7
5
r
пЂЅ 33
пЂЅ r
5
7
пЂЁ
3
n
пЂЅ
3
8
1/ 8
5/7
5
n
3 /7
пЂЅ m
8/3
Rules of Exponents
x
2
пѓ— x
3
пЂЅ x
2
n
2/3
пѓ—n
пЂ­1 / 2
пЂЅ n
3
пЂ«
2пЂ«3
пЂЅ x
пЂ­1
2
4
пЂЅ n
6
пЂ«
5
пЂ­3
6
1
пЂЅ n
6
Rules of Exponents
2x
пЂЁ8 c
4/5
4c
2
3/2
пѓ— 3x
пЂЅ 6x
5
4
пЂ­ 32 c 5
пЂ«
7
3
8
2
пЂЅ пЂ­ 32 c 10
пЂ«
15
23
10
пЂЅ пЂ­ 32 c 10
Rules of Exponents
x
5 /8
x
1/ 2
x
5
x
2
пЂЅ x
5
пЂЅ x8
пЂ­
5пЂ­2
1
5
2
пЂЅ x8
пЂЅ x
пЂ­
3
4
1
8
пЂЅ x8
Rules of Exponents
2 3
пЂЁm
3 /2 2/5
пЂЅ x
пЂЅ m
2 пѓ—3
3 2
пѓ—
2 5
пЂЅ x
3
пЂЅ m5
6
4
36
1
36
1/ 4
пЂЅ пЂЁ6
2 1/ 4
пЂЅ 6
2 1
пѓ—
1 42
1
пЂЅ 6
2
пЂЅ
6
6
27
1
27
1/ 6
пЂЅ пЂЁ3
3 1/ 6
пЂЅ 3
3 1
пѓ—
1 62
1
пЂЅ 3
2
пЂЅ
3
10
пЂЁa
2
b
8 1 / 10
пЂЅ a
2
a b
2пѓ—
1
10
8пѓ—
b
8
1
10
1
пЂЅ a b
5
4
5
пЂЅ пЂЁab
1
4 5
пЂЅ
5
ab
4
5
4
x
4
пѓ—
3
x
2
12
x5 пѓ— x3
пЂЅ x 15
пЂ«
2
10
22
15
пЂЅ x 15
пЂЅ
15
x
22
4 5
z
1
4
1
z5
пѓ¦ 5
пЂЅ пѓ§пѓ§ z
пѓЁ
1
пѓ¶4
пѓ·
пѓ·
пѓё
1
пЂЅ z 20
пЂЅ
20
z
Simplify. пЂЁ x
a)
x
3/8
y
3/4
b)
x
3/8
y
1/ 4
c) x
3/4
d) x
3/8
y
1/ 2
y
2/3
3/4
1/ 2
y
1/ 2
Slide 10- 23
Simplify. пЂЁ x
a)
x
3/8
y
3/4
b)
x
3/8
y
1/ 4
c) x
3/4
d) x
3/8
y
1/ 2
y
2/3
3/4
1/ 2
y
1/ 2