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# MTH55A_Lec-30_sec_7-2b_Rational_Exponents

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```Chabot Mathematics
В§7.2 Rational
Exponents
Bruce Mayer, PE
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Review В§ 7.2
MTH 55
вЂў В§7.2 в†’ HW-31
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Laws of Exponents
п‚§ For any real number a, any real number
b > 0, and any rational exponents m & n.
1.
2.
3.
m пЂ« n In multiplying, we can add exponents if the
bases are the same.
m
In dividing, we can subtract exponents if the
a
mпЂ­n
пЂЅa
bases are the same.
n
a n
m
m пѓ—n
To raise a power to a power, we can multiply
a
пЂЅa
the exponents.
a
m
пѓ—a
m
пЂЅa b
n
n
m m
a
пѓ¦aпѓ¶
пѓ§ пѓ· пЂЅ n
пѓЁbпѓё
b
Chabot College Mathematics
3
пЂЅa
5.
n
To raise a product to a power, we can raise
each factor to the power.
To raise a quotient to a power, raise both the
numerator & denominator to the power.
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
п‚§ Use the rules of exponents to simplify.
5/6
y
only positive exponents
пЂ­1 / 6
y
п‚§ SOLUTION
y
y
5/6
пЂ­1 / 6
пЂЅ y
5 / 6 пЂ­ ( пЂ­1 / 6 )
пЂЅ y
5 / 6 пЂ«1 / 6
пЂЅ y
Chabot College Mathematics
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Use the quotient for exponents.
(Subtract the exponents.)
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
п‚§ Use the Laws of Exponents to Simplify
a. 7
2/5
пѓ—7
1/ 5
b.
m
m
1/ 2
c.
1/ 4
пЂЁx
1 / 2 пЂ­1 / 3
y
п‚§ SOLUTION
a) 7
b)
2/5
m
m
Chabot College Mathematics
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пЂЅ7
пЂЅm
1 / 2 пЂ­1 / 4
1/ 2
1/ 4
2 / 5 пЂ«1 / 5
1/ 5
пѓ—7
пЂЅ7
пЂЅm
3/5
2 / 4 пЂ­1 / 4
пЂЅm
1/ 4
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
3/4
Example пѓ† Laws of Exponents
п‚§ Use the Laws of Exponents to Simplify
п‚§ SOLUTION c.
c)
пЂЁx
1 / 2 пЂ­1 / 3
y
3/4
пЂЁx
1 / 2 пЂ­1 / 3
y
пЂЅ x
пЂЅ x
3/4
(1 / 2)(3 / 4) ( пЂ­ 1 / 3)(3 / 4)
y
3 / 8 пЂ­1 / 4
y
пЂЅ
x
y
Chabot College Mathematics
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3/8
1/ 4
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
п‚§ Write with only positive exponents.
Assume that all variables are в‰Ґ 0
m1/4 nвЂ“6
mвЂ“8 n2/3
вЂ“3/4
Chabot College Mathematics
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=
(m1/4)вЂ“3/4 (nвЂ“6)вЂ“3/4
( mвЂ“8)вЂ“3/4 (n2/3)вЂ“3/4
=
mвЂ“3/16 n9/2
m6 nвЂ“1/2
=
mвЂ“3/16 вЂ“ 6 n9/2 вЂ“ (вЂ“1/2)
=
mвЂ“99/16 n5
=
n5
m99/16
Product to Power
Power-to-Power rule
Quotient rule
Definition of
Negative exponent
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
п‚§ Write with only positive exponents. All
variables represent positive numbers
x3/5(xвЂ“1/2 вЂ“ x3/4) = x3/5 В· xвЂ“1/2 вЂ“ x3/5 В· x3/4
= x3/5 + (вЂ“1/2) вЂ“ x3/5 + 3/4
Distributive property
Product rule
= x1/10 вЂ“ x27/20
п‚§ Do not make the common mistake of
multiplying exponents in the first step.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
that are powers.
п‚§ When these powers and the index
share a common factor, rational
exponents can be used to simplify
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
exponential expressions.
2. Use arithmetic and the laws of
exponents to simplify.
3. Convert back to radical notation when
appropriate.
п‚§ CAUTION: This procedure works only when all
expressions under radicals are nonnegative since
rational exponents are not defined otherwise. With this
assumption, no absolute-value signs will be needed.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
п‚§ Use rational exponents to simplify.
a.
8
x
b.
4
п‚§ SOLUTION
a.
8
x
4
пЂЅ x
4/8
пЂЅ x
1/ 2
пЂЅ
x
8
b.
8
4
a b
6
a b пЂЅ пЂЁa b
4
6
4
пЂЅa
4 /8
пЂЅa
1/ 2
пЂЅa
2/4
b
b
3/4
b
пЂЅ пЂЁa b
Chabot College Mathematics
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4
1/ 8
6 /8
2
пЂЅ
6
2
3/4
3
1/ 4
3
a bBruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
2
4
a)
x)
simplify. (3Do
п‚§ Use rational exponents to
not use exponents that are fractions in 9
пѓ¦3
2 пѓ¶
2
4
b ) пѓ§ xy z пѓ·
a)
(3 x )
пѓЁ
пѓ¦
2 пѓ¶
b ) пѓ§ 3 xy z пѓ·
пѓЁ
пѓё
2
a) 4 (3 x )
пѓё
9
c) 4
y
9
пѓ¦
2 пѓ¶
п‚§ SOLUTION
b ) 3 xy z
пѓ§
пѓЁ
c) 4
пѓ·
пѓё
2
2/4
a) 4 (3 x ) пЂЅ (3 x )
c) 4
y
1/ 2
пЂЅ (3 x )
Chabot College Mathematics
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пЂЅ
y
Convert to exponential notation
Simplify the exponent and
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
пѓ¦
2 пѓ¶
b ) пѓ§ 3 xy z пѓ·
пѓЁ
пѓё
2
4
(3 x )
Exponents
п‚§ SOLUTION
пѓ¦
2 пѓ¶
b ) пѓ§ 3 xy z пѓ·
пѓЁ
пѓё
9
c) 4
y
9
c)2 4 9 / 3y
пѓ¦3
2 пѓ¶
b) пѓ§ xy z пѓ· пЂЅ ( xy z )
пѓЁ
пѓё
2
3
3 6 3
пЂЅ ( xy z ) пЂЅ x y z
c) 4
y пЂЅ
Chabot College Mathematics
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4
y
1/ 2
пЂЁ
пЂЅ y
1/ 2
1/ 4
пЂЅ y
1/8
пЂЅ8 y
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
9
п‚§ Write a single
x пѓ—y
radical expression for x1 6 пѓ— y1 4
34
п‚§ SOLN
x
3/4
x
1/ 6
пѓ—y
5/8
пѓ—y
1/ 4
пЂЅ x
3 / 4 пЂ­1 / 6
пЂЅ x
9 /12 пЂ­ 2 /12
пЂЅ x
7 /12
пЂЅ x
14 / 24
пѓ—y
9 / 24
14
пѓ—y
9
пЂЅ
Chabot College Mathematics
14
24
x
пѓ—y
пѓ—y
58
5 / 8 пЂ­1 / 4
пѓ—y
5 / 8пЂ­2 / 8
3/8
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Rules of Exponents Summary
п‚§ Assume that no denominators are 0, that a and b
are real numbers, and that m and n are integers.
п‚§ Zero as an exponent:
a0 = 1, where a в‰ 0.
00 is indeterminate.
п‚§ Negative exponents: a пЂ­ n пЂЅ
1
a
n
,
п‚§ Product rule for exponents:
п‚§ Quotient rule for exponents:
1
a
пЂ­n
пЂЅa ,
n
a
b
a п‚ґa пЂЅ a
m
n
mпЂ­n
n
п‚§ Raising a power to a power:
пЂЁa
п‚§ Raising a product to a power:
пЂЁ ab пЂ© пЂЅ a b
п‚§ Raising a quotient to a power:
Chabot College Mathematics
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m
n
пЂЅa
n
a
b
n
mn
n
пЂЅ
пЂЅ
mпЂ« n
a п‚ё a пЂЅa
m
пЂ­n
a
n
b
n
n
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
b
a
n
Simplification GuideLines
п‚§ The GuideLines for Simplifying
expressions with Rational Exponents
1. No parentheses appear
2. No powers are raised to powers
3. Each Base Occurs only Once
4. No negative or zero exponents
appear
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Use Exponent Rules
п‚§ Rewrite all radicals as exponentials, and then
apply the rules for rational exponents. Leave
answers in exponential form. Assume c > 0
4
c
=
c1/4
c3/2
Convert to rational exponents.
=
c1/4 вЂ“ 3/2
Quotient rule
=
c1/4 вЂ“ 6/4
Write exponents with a common
denominator
=
cвЂ“5/4
c3
=
Chabot College Mathematics
17
1
c5/4
Definition of negative exponent
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
WhiteBoard Work
п‚§ Problems From В§7.2 Exercise Set
вЂў 58, 74, 78, 106, 110, 112, 132
п‚§
AmericaвЂ™s
Cup вЂњClass
RuleвЂќ 5.0
Formula
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
All Done for Today
Index
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Chabot Mathematics
Appendix
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu
вЂ“
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Graph y = |x|
6
п‚§ Make T-table
21
5
x
y = |x |
4
-6
6
3
-5
5
-4
4
-3
3
-2
2
-1
1
0
0
1
1
-2
2
2
-3
3
3
4
4
5
5
6
6
Chabot College Mathematics
y
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-4
-5
file = X Y _ P lo t_ 0 2 1 1 .xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
4
5
6
5
5
y
4
4
3
3
2
2
1
1
x
0
-1 0
-8
-6
-4
-2
0
2
4
6
-1
0
-3
-2
-1
0
1
2
3
4
5
-2
-1
-3
-2
-4
M 5 5 _ В§ J B e rla n d _ G ra p h s _ 0 8 0 6 .xls
-3
Chabot College Mathematics
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M 5 5 _ В§ J B e rla n d _ G ra p h s _ 0 8 0 6 .xls
-5
Bruce Mayer, PE
BMayer@ChabotCollege.edu вЂў MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
8
10
```
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