1. 5.1 Use Properties of Exponents

2. Properties of Exponents Explained 1
Product of Powers a a a n =
m+ n m
Example 5 3 5 5
3 + (-1) 5 -1 = = 2 = 25

3. Properties of Exponents Explained 2
Power of a Power
(a ) m n = a
m n
Example 2
(3 ) 3
3 2 3 6 =
729 = 3 =

4. Properties of Exponents Explained 3
Power of a Product
(ab) m
a b
m m =
Example 4
(2 3) =
1296 =
2 3

5. Properties of Exponents Explained 4
Negative Exponent 1 = -m
(a) a m 1 1
Example -2
(7) = = 7 49 2

6. Properties of Exponents Explained 5
Zero Exponent = 1
(a)
Example
(-89) = 1

7. Properties of Exponents Explained 6
Quotient of Powers a m
m - n = a a n
Example -3 6
-3 – (-6) 3 =
216 = = 6 6 -6 6

8. Properties of Exponents Explained 7
Quotient of Power m =
Example 2 = =

9. Power of a product property
EXAMPLE 1
Evaluate numerical expressions a.
(– )2
= (– 4)2 (25)2
Power of a product property =
Power of a power property =
= 16,384
Simplify and evaluate power. b.
115
118 –1
118
115 =
Negative exponent property
= 118 – 5
Quotient of powers property
= 113
= 1331
Simplify and evaluate power.

10. Substitute values. EXAMPLE 2 Use scientific notation in real life
Locusts
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?
SOLUTION
= 85,000,
Substitute values.

11. Write in scientific notation. Use multiplication properties.
EXAMPLE 2
Use scientific notation in real life
= ( )( )
Write in scientific
notation.
= ( )( )
Use multiplication
properties. =
Product of powers
property =
Write 10.2 in scientific
notation. =
Product of powers
property
The number of locusts is about ,
or about 102,000,000,000.
ANSWER

12. GUIDED PRACTICE
for Examples 1 and 2
Evaluate the expression. Tell which properties of
exponents you used.
(42 )3
4096 ; Power of a power property
ANSWER
2. (–8)(–8)3
ANSWER
4096 ; Product of a powers property 3.
2 3 9 8
ANSWER
; Power of a quotient property
729

13. GUIDED PRACTICE
for Examples 1 and 2
– 4 4. 2
ANSWER
; quotient of power property

14. Product of powers property
EXAMPLE 3
Simplify expressions
a. b–4b6b7
= b–
= b9
Product of powers property b.
r–2 –3 s3
( r –2 )–3
( s3 )–3 =
Power of a quotient property =
r 6
s–9
Power of a power property
= r6s9
Negative exponent property
c m4n –5
2n–5
= 8m4n –5 – (–5)
Quotient of powers property
= 8m4n0= 8m4
Zero exponent property

15. Power of a product property
EXAMPLE 4
Standardized Test Practice
SOLUTION
(x–3y3)2
x5y6 =
(x–3)2(y3)2
x5y6
Power of a product property
x –6y6
x5y6 =
Power of a power property

16. Quotient of powers property
EXAMPLE 4
Standardized Test Practice
= x –6 – 5y6 – 6
Quotient of powers property
= x–11y0
Simplify exponents.
= x–11 1
Zero exponent property = 1
x11
Negative exponent property
The correct answer is B.
ANSWER

17. GUIDED PRACTICE
for Examples 3, 4, and 5
Simplify the expression. Tell which properties of
exponents you used.
5. x–6x5 x3
ANSWER
x2 ; Product of powers property
(7y2z5)(y–4z–1)
7z4 y2
; Product of powers property, Negative exponent property
ANSWER

18. GUIDED PRACTICE
for Examples 3, 4, and 5 7.
s 3 2
t–4
s6t8
ANSWER
; Power of a power property, Negative exponent property 8.
x4y–
x3y6 x3
y24
; Quotient of powers property, Power of a Quotient property, Negative exponent property
ANSWER