2. VOCABULARY
53 = 5•5•5 = 125 “five to the third power”
exponent
base
product
factors
• Power – expression that represents repeated
multiplication of the same factor
• Exponent – the number of times the base is used as a
factor

3. 3 Properties of Products
Product of powers property
Power of a power property
Power of a product property

4. Product of Powers Property
a ×a = (a ×a)× ×a ×a) = a = a
2 3
(a 5 2+3
To multiply powers having the same base, add the
exponents.
Product of Powers: a ×a = a
m n m+n
5 ×5 = 5
6 3 6+3
=5 9

5. Example 1 Use the product of powers property
a. ( – 2 ) 3 • ( – 2 ) 5 = ( – 2) 3 + 5
= (– 2) 8
b. x4 • x3 = x4 + 3
= x7
c. 76 • 7 • 78 = 76 • 71 • 78
= 7 6+ 1 +8
= 715

6. Power of a Power Property
(a )
2 3
= (a 2 )×(a 2 )×(a 2 ) = (a ×a)×(a ×a)×(a ×a) = a 6 = a 2×3
To find a power of a power, multiply exponents.
Power of a Power: (a ) m n
=a m×
n
(3 ) 4 2
=3 =34×
2 8

7. Example 2 Use the power of a power property
a. ( 25 ) 3 = 25 • 3
= 215
b. [( – 6 ) 2 ] 5 = ( – 6 ) 2 • 5
10
= (–6 )
c. ( x 2 ) 4 = x2 • 4
= x8

8. Example 2 Use the power of a power property
d. [( y + 2 ) 6 ] 2 = ( y + 2 ) 6 • 2
= ( y + 2 )12

9. Power of a Product Property
( ab) = (ab)×(ab)×(ab) = (a ×a ×a)×(b ×b ×b) = a 3b 3
3
To find a power of a product, find the power of each
factor and multiply.
( ab)
m
Power of a Product: = a ×b
m m
( 23×17)
5
= 23 ×5
17 5

10. Example 3 Use the power of a product property
a. ( 24 • 13)8 = 248 • 138
b. ( 9xy )2 = ( 9 • x • y ) 2 = 92 • x2 • y2 = 81x2y2
c. ( – 4z ) 2 = (– 4 • z ) 2 = ( – 4 ) 2 • z2 = 16z2
d. – (4z ) 2 = – (4 • z ) 2 = –( 42 • z2 ) = –16z2

11. Summary of Exponent Properties
Product of Powers: a ×a = a
m n m+n
Power of a Power: (a )
m n
=a m×
n
( ab)
m
Power of a Product: = a ×b
m m

12. Example 4 Use all three properties
Simplify ( 2x 3 ) 2 • x4.
( 2x3 ) 2 • x4 = 22 • ( x 3 )2 • x4 Power of a product property
= 4 • x6 • x4 Power of a power property
= 4x10 Product of powers property