Mat 092 section 12.1 the power and product rules for exponents

Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Exponents and
Polynomials
12

1. Use exponents.
2. Use the product rule for exponents.
3. Use the rule (am)n
= amn.
4. Use the rule (ab)m = ambm.
5. Use the rule (a/b)m = am/bm.
6. Use combinations of the rules for
exponents.
7. Use the rules for exponents in a geometry
application.
Objectives
12.1 The Product Rule and Power Rules
for Exponents

Use Exponents
Exponent
(or Power)
Base 6 factors of 2
2 = 2 · 2 · 2 · 2 · 2 · 2
6
The exponential expression is 26, read “2 to the sixth power” or
simply “2 to the sixth.”
Example
Write 2 · 2 · 2 · 2 · 2 · 2 in exponential form and evaluate.
Since 2 occurs as a factor 6 times, the base is 2 and the
exponent is 6.
= 64

Evaluating Exponential Expressions
(a) 2 = 2 · 2 · 2 · 24
= 16
Base Exponent
2 4
(b) – 2 4
= –16
2 4
4= –1 · 2 = –1 · 2 · 2 · 2 · 2
(c) (–2)4
= 16
–2 4
= (–2) (–2) (–2) (–2)
Example Evaluate. Name the base
and the exponent.

Expression Base Exponent Example
Use Exponents
CAUTION
– an
(–a) n
In summary, and are not necessarily the same.– a n (–a) n
a n
(–a) n
– 52
(– 5)2
= – ( 5 · 5 ) = – 25
= (–5) (–5) = 25

Use the Product Rule for Exponents
Product Rule for Exponents
For any positive integers m and n, a m · a n = a m + n
(Keep the same base and add the exponents.)
53 · 54
Example:
( 5 · 5 · 5 ) = 5 3 + 4
( 5 · 5 · 5 · 5 ) = 57

CAUTION
Do not multiply the bases when using the product rule.
Keep the same base and add the exponents.
Example: 3 4 7 7
5 5 5 not 25 . 

Using the Product Rule
= 75 + 85 8(a) 7 · 7 = 713
= y1 + 55(c) y · y = y65= y · y1
= (– 2)4 + 54 5(b) (–2) (–2) = (– 2)9
Example
Use the product rule for exponents to simplify, if possible.

= n 2 + 42 4(d) n · n = n 6
The product rule does not apply
because the bases are different.
2 2
(e) 3 · 2
The product rule does not apply
because it is a sum, not a product.
3 2
(f) 2 + 2
= 8 + 4 = 12
= 9 · 4 = 36
Example (cont)
Use the product rule for exponents to simplify, if possible.

Add the exponents.
3 6
Multiply 5 4 .y y
Example
3 6
5 4y y
  3 6
5 4 y y  
3 6
20y 
 Multiply; product rule
Commutative and associative properties
9
20y

CAUTION
Be sure you understand the difference between adding and
multiplying exponential expressions. For example,
2 2
7k + 3k
22
= ( 7 + 3 )k = 10k ,
42 + 2
= ( 7 · 3 )k = 21k .
2 2
7k 3kbut,
Add.
Multiply.

Use the Rule (am)n
= amn
Power Rule (a) for Exponents
For any positive integers m and n,
m n
( a )
Example:
3 2
( 4 )
3 · 2
= 4
(Raise a power to a power by multiplying exponents.)
6
= 4 .
m n
= a .

Use the Rule (am)n
= amn
Example
Use power rule (a) for exponents to simplify.
(a) ( 3 )2 5
= 32 · 5 = 3 10
(b) ( 4 )8 6
= 48 · 6 = 4 48
(c) ( n )7 3
= n 7 · 3 = n 21
(d) ( 2 )5 8 = 2 5 · 8 = 2 40

Use the Rule (ab)m = ambm
Power Rule (b) for Exponents
For any positive integer m,
m
( ab )
m m
= a b .
Example:
3
( 5h )
(Raise a product to a power by raising each factor to the power.)
3
= 5 h .3

Example
Use power rule (b) for exponents to simplify.
(a) ( 2abc ) 4 = 2 a b c4 4 4 4
= 16 a b c4 4 4
Power rule (b)
= 5 ( x y )2 6 Power rule (b)(b) 5 ( x y ) 23
= 5x y2 6

Example (cont)
Power rule (b)
(c) 7 ( 2m n p )75 3
= 7 [ 2 ( m ) ( n ) ( p ) ]3 5 3 71 3 3
Power rule (a)= 7 [ 8 m n p ]3 15 21
= 56 m n p3 15 21

Example (cont)
Power rule (b)
(d) ( –3 ) 54
Power rule (a)
= ( – 1 · 3 ) 54
= ( – 1 ) ( 3 ) 545
= – 1 · 3 20
= – 3 20
– a = – 1 · a

CAUTION
Power rule (b) does not apply to a sum:
   
2 22 2 2 2
5 5 , but 5 5 .y y y y  

Use the Rule (a/b)m = am/bm
Power Rule (c) for Exponents
For any positive integer m,
(Raise a quotient to a power by raising both the numerator
and the denominator to the power.)
ma
b
ma
bm
=
Example:
23
4
23
4 2
=
( b ≠ 0 ).

Use the Rule (a/b)m = am/bm
Example
Use power rule (c) for exponents to simplify.
42
5
42
5 4
=(a)
16
625
=
7x
y
7x
y 7
=(b) y ≠ 0

Rules for Exponents
For any positive integers m and n: Examples
Product rule am · an = am + n
34 · 35 = 39
Power rules
m n
( a )
m n
= a(a) 4 5
( 2 )
20
= 2
m m m
(b) ( ab ) = a b
3
( 4k )
3
= 4 k
3
(c)
ma
b
ma
bm
=
( b ≠ 0 ).
34
7
=
34
7 3

Use Combinations of the Rules for Exponents
Example
Simplify each expression.
Power rule (c)
Multiply fractions.
(a) · 35
23
4
= ·
23
42
53
1
=
23 ·
4 ·2
53
1
Product rule=
3
42
7

Example (cont)
Product rule
(b) ( 7m n ) ( 7m n )22 3 5
= ( 7m n )2 8
Power rule (b)= 7 m n16 88

Example (cont)
(c) ( 2x y ) ( 2 x y ) 1063 7 4
= ( 2 ) ( x ) ( y ) · ( 2 ) ( x ) ( y )10 7 46 3 6 6 10 10
= 2 · x · y · 2 · x · y10 70 406 18 6 Power rule (a)
= 2 · 2 · x · x · y · y70 6 406 10 18 Commutative and
associative properties
= 2 · x · y16 88 46 Product rule
Power rule (b)

Example (cont)
Power rule (b)
(d) ( – g h ) ( – g h )3 2 5 32
= ( – 1 g h ) ( – 1 g h )3 2 5 32
= ( – 1 ) ( g ) ( h ) ( – 1 ) ( g ) ( h )2 26 3 6 15
Product rule= ( – 1 ) ( g ) ( h )5 12 17
= – 1 g h12 17

Use the Rules for Exponents in a Geometry Application
Example
Find a polynomial that represents the area of the geometric figure.
(a) Use the formula for the area of a rectangle, A = LW.
A = ( 4x )( 2x )3 2
A = 8x 5
Product rule
4x 3
2x 2

Use the Rules for Exponents in a Geometry Application
Example
Find a polynomial that represents the area of the geometric figure.
A = 12n9
Product rule
8n5
3n4
A = ( 3n ) ( 8n )4 51
2
(b) Use the formula for the area of a triangle, A = LW.1
2
A = ( 24n )91
2

Mat 092 section 12.1 the power and product rules for exponents

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