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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.6 §1.6 ExponentExponentPropertiesProperties
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §1.5 → (Word) Problem Solving
Any QUESTIONS About HomeWork• §1.5 → HW-01
1.5 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Exponent PRODUCT RuleExponent PRODUCT Rule
For any number a and any positive integers m and n,
nmnm aaa In other Words:
To MULTIPLY powers with the same base, keep the base and ADD the exponents
Exponent
Base
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Quick Test of Product RuleQuick Test of Product Rule
nmnm aaa Test 532
?32 3333
24327933 32
243279333333333335
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product Rule Product Rule
Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)
a) x3 x5 b) 62 67 63
c) (x + y)6(x + y)9 d) (w3z4)(w3z7)
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product Rule Product Rule
Solution a) x3 x5 = x3+5 Adding exponents
= x8
Solution b) 62 67 63 = 62+7+3
= 612
Solution c) (x + y)6(x + y)9 = (x + y)6+9
= (x + y)15
Solution d) (w3z4)(w3z7) = w3z4w3z7
= w3w3z4z7
= w6z11
Base is x
Base is 6
Base is (x + y)
TWO Bases: w & z
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Bruce Mayer, PE Chabot College Mathematics
Exponent QUOTIENT RuleExponent QUOTIENT Rule
For any nonzero number a and any positive integers m & n for which m > n, nm
n
m
aa
a In other Words:
To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator
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Bruce Mayer, PE Chabot College Mathematics
Quick Test of Quotient RuleQuick Test of Quotient Rule
Test246
?
4
6
555
5
nmn
m
aa
a
5555
555555
5
54
6
462 5555
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Bruce Mayer, PE Chabot College Mathematics
Example Example Quotient Rule Quotient Rule
Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)
• a) b)
• c) d)
9
3
x
x
7
3
8
8
14
6
(6 )
(6 )
y
y
7 9
3
6
4
r t
r t
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Bruce Mayer, PE Chabot College Mathematics
Example Example Quotient Rule Quotient Rule
Solution a)9
9 33
xx
x 6x
Solution b)7
7 33
88
8 48
Solution c)14
14 6 86
(6 )(6 ) (6 )
(6 )
yy y
y
Solution d)7 9 7 9
3 3
66
4 4
r t r t
tr t r
7 3 9 41 8
4
36
2r t r t
Base is x
Base is 8
Base is (6y)
TWO Bases: r & t
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Bruce Mayer, PE Chabot College Mathematics
The Exponent ZeroThe Exponent Zero
For any number a where a ≠ 0 10 a
In other Words: Any nonzero number raised to the 0 power is 1• Remember the base can be
ANY Number–0.00073, 19.19, −86, 1000000, anything
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Bruce Mayer, PE Chabot College Mathematics
Example Example The Exponent Zero The Exponent Zero Simplify: a) 12450 b) (−3)0
c) (4w)0 d) (−1)80 e) −80
Solutionsa) 12450 = 1
b) (−3)0 = 1
c) (4w)0 = 1, for any w 0.
d) (−1)80 = (−1)1 = −1
e) −80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80 = (−1)1 = −1
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Bruce Mayer, PE Chabot College Mathematics
The POWER RuleThe POWER Rule
For any number a and any whole numbers m and n
nmnm aa In other Words:
To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged
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Bruce Mayer, PE Chabot College Mathematics
Quick Test of Power RuleQuick Test of Power Rule
Test 632?32 777
494949497 332
nmnm aa
67777777777777 327
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Bruce Mayer, PE Chabot College Mathematics
Example Example Power Rule Power Rule
Simplify: a) (x3)4 b) (42)8
Solution a) (x3)4 = x34
= x12
Solution b) (42)8 = 428
= 416
Base is x
Base is 4
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Raising a Product to a PowerRaising a Product to a Power
For any numbers a and b and any whole number n,
nnn baba In other Words:
To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER
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Bruce Mayer, PE Chabot College Mathematics
Quick Test of Product to PowerQuick Test of Product to Power
Test 33?
3 112112
1064822222222112 33
1064813318112 33
nnn baba
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product to Power Product to Power
Simplify: a) (3x)4 b) (−2x3)2
c) (a2b3)7(a4b5) Solutions
a) (3x)4 = 34x4 = 81x4
b) (−2x3)2 = (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6
c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5
= a14b21a4b5 Multiplying exponents
= a18b26 Adding exponents
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Raising a Quotient to a PowerRaising a Quotient to a Power
For any real numbers a and b, b ≠ 0, and any whole number n
n
nn
b
a
b
a
In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power
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Quick Test of Quotient to PowerQuick Test of Quotient to Power
Test
3
3?3
7
5
7
5
3
33
7
5
777
555
343
125
7
5
7
5
7
5
7
5
n
nn
b
a
b
a
3
33
7
5
7
5
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Bruce Mayer, PE Chabot College Mathematics
Example Example Quotient to a Power Quotient to a Power
Simplify: a) b) c) 3
4
w
4
5
3
b
25
4
2a
b
Solution a)3 33
34 644
w w w
4 4
5 45
3 3
( )b b 5 4 20
81 81
b b Solution b)
25 5
4 4
2
2
2 (2 )
( )
a a
b b
2 5 2 10
4 2 8
2 ( ) 4a a
b b Solution c)
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Negative Exponents Negative Exponents
Integers as Negative Exponents
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Negative ExponentsNegative Exponents
For any real number a that is nonzero and any integer n
nn
aa
1
The numbers a−n and an are thus RECIPROCALS of each other
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Bruce Mayer, PE Chabot College Mathematics
Example Example Negative Exponents Negative Exponents
Express using POSITIVE exponents, and, if possible, simplify.
a) m–5 b) 5–2 c) (−4)−2 d) xy–1
SOLUTION
a) m–5 =
b) 5–2 =
5
1
m
2
1 1
255
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Bruce Mayer, PE Chabot College Mathematics
Example Example Negative Exponents Negative Exponents
Express using POSITIVE exponents, and, if possible, simplify.
a) m–5 b) 5–2 c) (−4)−2 d) xy−1
SOLUTION
c) (−4)−2 =
d) xy–1 =
2
1 1 1
( 4)( 4) 16( 4)
1
1 1 xx x
y yy
• Remember PEMDAS
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More ExamplesMore Examples
Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3
d) e) f)
Solution
a)
5 3w w5
6
a
a
9
1
b
7
6
w
z
5 3w w 5 ( 3 2)w w
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Bruce Mayer, PE Chabot College Mathematics
More ExamplesMore Examples
Solution
b) (x−4)−3 = x(−4)(−3) = x12
c) (3a2b−4)3 = 33(a2)3(b−4)3
= 27 a6b−12 =
d)
6
12
27a
b515 ( )
66a
a a aa
e) ( 9 99
)1b b
b
f) 7 6
7 66 6 7 7
1 1w zw z
z z w w
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Bruce Mayer, PE Chabot College Mathematics
Factors & Negative ExponentsFactors & Negative Exponents
For any nonzero real numbers a and b and any integers m and n
n
m
m
n
a
b
b
a
A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed
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Bruce Mayer, PE Chabot College Mathematics
Examples Examples Flippers Flippers
Simplify 6
3 4
20
4
x
y z
SOLUTION We can move the negative factors to
the other side of the fraction bar if we change the sign of each exponent.
3
4
34
6
6
20 5
4
x z
y z y z
6x
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Reciprocals & Negative ExponentsReciprocals & Negative Exponents
For any nonzero real numbers a and b and any integer n
nn
a
b
b
a
Any base to a power is equal to the reciprocal of the base raised to the opposite power
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Examples Examples Flippers Flippers
Simplify
SOLUTION
24
3
a
b
4
4
223
3
ba
ab
2
4 2
(3 )
( )
b
a
2 2
8
2
8
3 9b b
a a
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Bruce Mayer, PE Chabot College Mathematics
Summary – Exponent PropertiesSummary – Exponent Properties1 as an exponent a1 = a
0 as an exponent a0 = 1
Negative Exponents(flippers)
The Product Rule
The Quotient Rule
The Power Rule (am)n = amn
The Product to a Power Rule
(ab)n = anbn
The Quotient to a Power Rule
.n n
n
a a
b b
.m
m nn
aa
a
.m n m na a a
1, ,
n nn mn
n m n
a b a ba
b aa b a
This sum
mary assum
es that no denom
inators are 0 and that 00 is not
considered. For any integers m
and n
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §1.6 Exercise Set• 14, 24, 52, 70, 84, 92, 112, 130
Base & Exponent →Which is Which?
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All Done for TodayAll Done for Today
AstronomicalUnit(AU)
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22