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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
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Review В§ 7.2
MTH 55
 Any QUESTIONS About
• §7.2 → Radical Functions
 Any QUESTIONS About HomeWork
• §7.2 → HW-31
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Laws of Exponents
 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
пЂЁ пЂ©
4. пЂЁ a b пЂ©
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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
 Use the rules of exponents to simplify.
5/6
Write the answer with
y
only positive exponents
пЂ­1 / 6
y
 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
4
Use the quotient for exponents.
(Subtract the exponents.)
Rewrite the subtraction as addition.
Add the exponents.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
 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
 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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
пЂ©
3/4
Example пѓ† Laws of Exponents
 Use the Laws of Exponents to Simplify
 SOLUTION c.
c)
пЂЁx
1 / 2 пЂ­1 / 3
y
пЂ©
3/4
пЂЁx
1 / 2 пЂ­1 / 3
y
пЂЅ x
пЂЅ x
пЂ©
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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
 Write with only positive exponents.
Assume that all variables are ≥ 0
m1/4 n–6
m–8 n2/3
–3/4
Chabot College Mathematics
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=
(m1/4)–3/4 (n–6)–3/4
( m–8)–3/4 (n2/3)–3/4
=
m–3/16 n9/2
m6 n–1/2
=
m–3/16 – 6 n9/2 – (–1/2)
=
m–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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Laws of Exponents
 Write with only positive exponents. All
variables represent positive numbers
x3/5(x–1/2 – x3/4) = x3/5 · x–1/2 – x3/5 · x3/4
= x3/5 + (–1/2) – x3/5 + 3/4
Distributive property
Product rule
= x1/10 – x27/20
 Do not make the common mistake of
multiplying exponents in the first step.
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Simplifying Radical Expressions
 Many radical expressions contain
radicands or factors of radicands
that are powers.
 When these powers and the index
share a common factor, rational
exponents can be used to simplify
the radical expression.
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Simplifying Radical Expressions
1. Convert radical expressions to
exponential expressions.
2. Use arithmetic and the laws of
exponents to simplify.
3. Convert back to radical notation when
appropriate.
 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
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Radical Exponents
 Use rational exponents to simplify.
a.
8
x
b.
4
 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
пЂ©
1/ 8
6 /8
2
пЂЅ
6
2
3/4
3
пЂ©
1/ 4
3
a bBruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Radical Exponents
2
4
a)
x)
simplify. (3Do
 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 )
the final answer.
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¶
 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
12
пЂЅ
y
Convert to exponential notation
Simplify the exponent and
3 x return to radical notation
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
пѓ¦
2 пѓ¶
b ) пѓ§ 3 xy z пѓ·
пѓЁ
пѓё
2
4
(3 x )
Example пѓ† a)Radical
Exponents
 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
пЂ©
1/ 4
пЂЅ y
1/8
пЂЅ8 y
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
9
Example пѓ† Radical Exponents
 Write a single
x пѓ—y
radical expression for x1 6 пѓ— y1 4
34
 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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Rules of Exponents Summary
 Assume that no denominators are 0, that a and b
are real numbers, and that m and n are integers.
 Zero as an exponent:
a0 = 1, where a в‰ 0.
00 is indeterminate.
 Negative exponents: a  n 
1
a
n
,
 Product rule for exponents:
 Quotient rule for exponents:
1
a
пЂ­n
пЂЁ пЂ©
пЂЅa ,
n
a
b
a п‚ґa пЂЅ a
m
n
mпЂ­n
n
 Raising a power to a power:
пЂЁa
 Raising a product to a power:
пЂЁ ab пЂ© пЂЅ a b
 Raising a quotient to a power:
Chabot College Mathematics
15
m
пЂ©
n
пЂЅa
n
пЂЁ пЂ©
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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
пЂЁ пЂ©
b
a
n
Simplification GuideLines
 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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Example пѓ† Use Exponent Rules
 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 – 3/2
Quotient rule
=
c1/4 – 6/4
Write exponents with a common
denominator
=
c–5/4
c3
=
Chabot College Mathematics
17
1
c5/4
Definition of negative exponent
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
WhiteBoard Work
 Problems From §7.2 Exercise Set
• 58, 74, 78, 106, 110, 112, 132

America’s
Cup “Class
Rule” 5.0
Formula
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
All Done for Today
Radical
Index
Radicand
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Chabot Mathematics
Appendix
r пЂ­ s п‚є пЂЁr пЂ­ s пЂ©пЂЁr пЂ« s пЂ©
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
Graph y = |x|
6
 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 • 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 • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt
8
10
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