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5.1 Use Properties of Exponents
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Properties of Exponents Explained 1Product of Powers
a am n· =a m+ n
Example
5 555 25
3 -1 3 + (-1)=· = 2=
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Properties of Exponents Explained 2Power of a Power
(a )
m n =a m n
Example
(3 ) 33 729
3 2 3 2=
·
=6=·
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Properties of Exponents Explained 3Power of a Product
(ab)m=a
b
m m
Example
(2 3) 2 3 1296
4 4 4= =
··
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Properties of Exponents Explained 4Negative Exponent
(a)-m= 1
m Example
(7) 7 49-2
2
= =
a1 1
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Properties of Exponents Explained 5Zero Exponent
(a) 0 =1
Example
(-89)0
=1
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Properties of Exponents Explained 6Quotient of Powers
aa
m
n =am - n
Example
6 666
216
-3
-6
-3 – (-6)
= =3=
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Properties of Exponents Explained 7
Quotient of Power
m
=Example
2
= =
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EXAMPLE 1 Evaluate numerical expressions
a. (–4 25)2
= 16 25 2
Power of a product property
Power of a power property
Simplify and evaluate power.
= 118 – 5
Negative exponent property
Quotient of powers property
Simplify and evaluate power.= 113 = 1331
= 16 210 = 16,384
b. 115
118
–1
= (– 4)2 (25)2
118
115=
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EXAMPLE 2 Use scientific notation in real life
A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts are in such a swarm?
Locusts
SOLUTION
Substitute values.= 85,000,000 1200
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EXAMPLE 2 Use scientific notation in real life
Write in scientificnotation.
= (8.5 107)(1.2 103)
Use multiplication properties.
Product of powers property
Write 10.2 in scientificnotation.
Product of powersproperty
= 10.2 1010
= 1.02 1011
= (8.5 1.2)(107 103)
= 1.02 101 1010
The number of locusts is about 1.02 1011, or about 102,000,000,000.
ANSWER
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GUIDED PRACTICE for Examples 1 and 2
Evaluate the expression. Tell which properties of exponents you used.
1. (42 )3
4096 ; Power of a power property
2. (–8)(–8)3
4096 ; Product of a powers property
3. 2 3
9
8729 ; Power of a quotient property
ANSWER
ANSWER
ANSWER
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GUIDED PRACTICE for Examples 1 and 2
6 10 – 4
9 1074.
; quotient of power property 2
3 1011ANSWER
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EXAMPLE 3 Simplify expressions
a. b–4b6b7 Product of powers property
b. r–2 –3
s3
( r –2 )–3
( s3 )–3 = Power of a quotient property
= r 6
s–9Power of a power property
= r6s9 Negative exponent property
c. 16m4n –5
2n–5= 8m4n –5 – (–5) Quotient of powers property
= 8m4n0= 8m4 Zero exponent property
= b–4 + 6 + 7 = b9
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EXAMPLE 4 Standardized Test Practice
SOLUTION
(x–3y3)2
x5y6=
(x–3)2(y3)2
x5y6
x –6y6
x5y6=
Power of a product property
Power of a power property
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EXAMPLE 4
= x –6 – 5y6 – 6 Quotient of powers property
= x–11y0 Simplify exponents.
Zero exponent property
= 1x11 Negative exponent property
= x–11 1
The correct answer is B. ANSWER
Standardized Test Practice
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GUIDED PRACTICE for Examples 3, 4, and 5
Simplify the expression. Tell which properties ofexponents you used.
5. x–6x5 x3
x2 ; Product of powers property
6. (7y2z5)(y–4z–1)
7z4
y2
; Product of powers property, Negative exponent property
ANSWER
ANSWER
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GUIDED PRACTICE for Examples 3, 4, and 5
7. s 3 2
t–4
s6t8; Power of a power property, Negative exponent property
8. x4y–2 3
x3y6
; Quotient of powers property, Power of a Quotient property, Negative exponent property
x3
y24 ANSWER
ANSWER