Chapter 11
Adv. Math +
What do you do with exponents when
you multiply the bases?
add them!
What do you do with exponents when
you divide the bases?
subtract denominator’s from numerator’s
What is anything to the zero power?
one!
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2
Section 11.1
Adv. Math +
!  Use
the properties of exponents
!  Evaluate
and simplify
expressions containing rational
exponents
!  Solve
equations containing
rational exponents
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4
1. am · an = am+n
Product Property
2. am = am-n
an
Quotient Property
3. (am)n = am·n
Power Property
4. (ab)m = ambm
Power of a Product
5. (a/b)m = am/bm
Power of a Quotient
6. a-m = 1
m
a
Negative Exponent
7. a0 = 1 Zero Exponent
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5
Square Roots 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…
Cube Roots
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000…
Fourth Roots
1, 16, 81, 256, 625, 1296…
Fifth Roots
1, 32, 243, 1024, 3125…
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6
It is common practice to write
our answers using
POSTIVE exponents ONLY!
**(unless otherwise stated)**
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11
1. 50 · 3-4
1
34
1
81
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2. (23)4
212
3. (¾)2
32
42
9
16
12
4. -5x4y-10
10x3y-9
-5x4y9
3
10
10x y
-1x
2y
5. (73·3-2)-2·35
7-6 ·34 · 35
39
6
7
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13
6. -16x4yz-5 -2x-5y3z-12
-16x4x5y z12
-2 y3 z5
8 x9 z7
y2
7 . (3/5)-3
3-3/5-3
53/33
125/27
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14
exponent
index
n
x
exponent
m
base
“the nth root of x to
the mth power”
Adv. Math +
base
x
m
n
index
“x to the m over n
power”
15
½
x
1. √x = 2/3
3
2
2. √y = y 3.
8√52
4.
4√95
5.
5√y
Adv. Math +
= 5
=
2/8
=
1/4
5
5/4
9
– 2 = (y –
2
2/5
2)
16
1. 3
2/9
=
2. 4
3/10
3. 6
2/3
4. x
9/4
9√32
10√43
=
=
3√62
=
4√x9
5. (3x – 4)3/2 = 2√3x – 4
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3
17
1. am · an = am+n
Product Property
2. am = am-n
an
Quotient Property
3. (am)n = am·n
Power Property
4. (ab)m = ambm
Power of a Product
5. (a/b)m = am/bm
Power of a Quotient
6. a-m = 1
m
a
Negative Exponent
7. a0 = 1 Zero Exponent
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1. 5
½
∙
¼
5
½
+
¼
5
¾
5
Adv. Math +
2. (8
1/2·2
8
½
1/3
2
5 )
∙
·
1/3·2
5
1
2/3
8 ·5
19
3. (2 4 ∙3 4) -1/4
4. 7/ 7 1/3 2 4 ·-1/4 · 3 4·-1/4 7 1 -1/3
2 -1 · 3 -1
3/3 -1/3
7
(2 · 3)-1
(6)-1
7 2/3 1
6
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5. (27 1/3 ∙ 6 1/4) 2
6. 6 / 6 3/4 271/3 · 2 · 6 ¼ · 2 61 – ¾ 27 2/3 · 6 1/2
6 1/4
3√27 2 ∙ √6
32 ∙√6
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9√6
21
1. 4√53 = 53/4 = 5^(3/4) ≈ 3.34
2. 3√34 = 34/3 = 3^(4/3) ≈ 4.33
3.
5√723
Adv. Math +
3/5 =
72
72^(3/5) ≈ 13.01
=
22
Square Roots 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…
Cube Roots
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000…
Fourth Roots
1, 16, 81, 256, 625, 1296…
Fifth Roots
1, 32, 243, 1024, 3125…
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2.
1. 3√250y7
Cube Roots:
1,8,27,64,125,216,343
3√125∙2∙y7
5y2 3√2y
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4
x
y8
=
4
4
x
y
=
4
4
4
8
2
=
x
4
y
8
x
y
2
25
3. 3√81z5
4. 4√162x9
Cube Roots:
Fourth Roots
1,8,27,64,125,216,343 1,16,81,256
3√81z5 =
3√27∙3∙z5
=
3z 3√3z2
Adv. Math +
4√162x9
=
4√81∙2∙x9
=
3x2 4√2x
26
5. 3√56x3z5
6. 4√48x12y15
Cube Roots:
Fourth Roots
1,8,27,64,125,216,343 1,16,81,256
3√8·7x3z5 =
4√48x12y15=
2xz 3√7z2
4√16∙3x12y15
2x3y3 4√3y3
=
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3
4
1. 734 = x + 5
729 = x
4
3
3
4
3 4
4 3
(729) = (x )
x ≈ 6561
Adv. Math +
4
5
= x −9
2. 616
625 = x
5
4
4
5
4 5
5 4
(625) = (x )
x = 3125
28
!  List
the seven properties of
exponents.
(use x and y as bases and p and q as exponents)
!  What
should you do with a
negative exponent?
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p. 700
4-18 all, 20, 21, 22, 24,
25, 28, 29, 37-59 odd,
63-67 all
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