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Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Objectives
2
6
5
3
4
Integer Exponents and Scientific Notation
Use the product rule for exponents.
Define 0 and negative exponents.
Use the quotient rule for exponents.
Use the power rules for exponents.
Simplify exponential expressions.
Use the rules for exponents with scientific notation.
5.1
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
We use exponents to write products of repeated factors. For example,
25 is defined as 2 • 2 • 2 • 2 • 2 = 32.
The number 5, the exponent, shows that the base 2 appears as a factor five times. The quantity 25 is called an exponential or a power.We read 25 as “2 to the fifth power” or “2 to the fifth.”
Integer Exponents and Scientific Notation
Slide 5.1- 3
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the product rule for exponents.
Objective 1
Slide 5.1- 4
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Product Rule for ExponentsIf m and n are natural numbers and a is any real number, then
am • an = am + n.
That is, when multiplying powers of like bases, keep the same base and add the exponents.
Slide 5.1- 5
Use the product rule for exponents.
Be careful not to multiply the bases. Keep the same base and add the exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Apply the product rule, if possible, in each case.
m8 • m6
m5 • p4
(–5p4) (–9p5)
(–3x2y3) (7xy4)
= m8+6 = m14
Cannot be simplified further because the bases m and pare not the same. The product rule does not apply.
= 45p9= (–5)(–9)(p4p5) = 45p4+5
= –21x3y7 = (–3)(7) x2xy3y4 = –21x2+1y3+4
Slide 5.1- 6
CLASSROOM EXAMPLE 1
Using the Product Rule for Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Define 0 and negative exponents.
Objective 2
Slide 5.1- 7
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Zero ExponentIf a is any nonzero real number, then
a0 = 1.
Slide 5.1- 8
Define 0 and negative exponents.
The expression 0 0 is undefined.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate.
290
(–29)0
–290
80 – 150
= 1
= 1
= – (290) = –1
= 1 – 1 = 0
Slide 5.1- 9
CLASSROOM EXAMPLE 2
Using 0 as an Exponent
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Negative Exponent
For any natural number n and any nonzero real number a,
1− = .nn
aa
A negative exponent does not indicate a negative nu mber; negative exponents lead to reciprocals.
22
1 13 Not negative
3 9−
−= = 22
1 13 Negati
9v
3e−
−=− −=−
Slide 5.1- 10
Define 0 and negative exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write with only positive exponents.
6-5
(2x)-4 , x ≠ 0
–7p-4, p ≠ 0
Evaluate 4-1 – 2-1.
5
1
6=
( )4
1, 0
2x
x= ≠
4
17
p
= −
4
7, 0p
p= − ≠
1 1
4 2= −
1 2
4 4= − 1
4= −
Slide 5.1- 11
CLASSROOM EXAMPLE 3
Using Negative Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate.
3
1
4−
3
1319
=
3
114
=3
11
4= ÷
341
1= ⋅ 34= 64=
3
1
3
9
−
− 3
1 1
3 9= ÷
3
1 9
3 1= ⋅ 1 9
27 1= ⋅ 9
27= 1
3=
Slide 5.1- 12
CLASSROOM EXAMPLE 4
Using Negative Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Special Rules for Negative Exponents
If a ≠ 0 and b ≠ 0 , then
and1− = n
na
a
−
− = .n m
m n
a bb a
Slide 5.1- 13
Define 0 and negative exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the quotient rule for exponents.
Objective 3
Slide 5.1- 14
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Quotient Rule for Exponents
If a is any nonzero real number and m and n are integers, then
That is, when dividing powers of like bases, keep the same base and subtract the exponent of the denominator from the exponent of the numerator.
−= .m
m nn
aa
a
Slide 5.1- 15
Use the quotient rule for exponents.
Be careful when working with quotients that involve negative exponents in the denominator. Write the numerator exponent, then a subtraction symbol, and then the denominator exponent. Use parentheses.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Apply the quotient rule, if possible, and write each result with only positive exponents.
8
13
m
m
6
8
5
5
−
−
8 13 55
1, 0m m m
m− −= = = ≠
6 ( 8) 6 8 25 5 5 , or 25− − − − += = =
3
5, 0
xy
y≠ Cannot be simplified because the bases x and y are
different. The quotient rule does not apply.
Slide 5.1- 16
CLASSROOM EXAMPLE 5
Using the Quotient Rule for Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the power rules for exponents.
Objective 4
Slide 5.1- 17
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Power Rule for ExponentsIf a and b are real numbers and m and n are integers, then
and
That is,a) To raise a power to a power, multiply exponents.b) To raise a product to a power, raise each factor to that power.c) To raise a quotient to a power, raise the numerator and the
denominator to that power.
( ) ( )
0
= =
= ≠
a) , b) ,
c) ( ).
n mm mn m m
m m
m
a a ab a b
a ab
b b
Slide 5.1- 18
Use the power rules for exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify, using the power rules.
( )45r ( )45r=
( )253y−
33
4
5 4r= � 20r=
( ) ( )22 53 y= − 5 29y= � 109y=
3
3
3
4= 27
64=
Slide 5.1- 19
CLASSROOM EXAMPLE 6
Using the Power Rules for Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Special Rules for Negative Exponents, Continued
If a ≠ 0 and b ≠ 0 and n is an integer, then
and
That is, any nonzero number raised to the negative nth power is equal to the reciprocal of that number raised to the nth power.
1− =
nna
a
− =
.n n
a bb a
Slide 5.1- 20
Use the power rules for exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write with only positive exponents and then evaluate.
42
3
−
43
2 =
4
4
3
2= 81
16=
Slide 5.1- 21
CLASSROOM EXAMPLE 7
Using Negative Exponents with Fractions
Solution:
51
, 02
xx
− ≠
52
1
x =
52x= 532x=
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Definition and Rules for Exponents
For all integers m and n and all real numbers a and b, the following rules apply.
Product Rule
Quotient Rule
Zero Exponent
+⋅ =m n m na a a
( )0−= ≠ m
m nn
aa a
a
( )0 1 0= ≠ a a
Slide 5.1- 22
Use the power rules for exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Definition and Rules for Exponents, Continued
Negative Exponent
Power Rules
Special Rules
10− = ≠ ( )n
na a
a
( ) = nm mna a
( )0 = ≠
m m
m
a ab
b b
( ) =m m mab a b
( )10− = ≠ n
na a
a
( )10− = ≠
n
na aa
( )0−
− = ≠ ,n m
m n
a ba b
b a
( )0−
= ≠
,n n
a ba b
b a
Slide 5.1- 23
Use the power rules for exponents.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify exponential expressions.
Objective 5
Slide 5.1- 24
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Simplify. Assume that all variables represent nonzero real numbers.
( ) 524−
4 6 8x x x− −� �
2( 5)4 −= 104−= 10
1
4=
( )6 84x + − +−= 2x−= 2
1
x=
( ) 22
3
m n
m n
−
−
( ) 22 2
3
m n
m n
− −
−=4 2
3
m n
m n
− −
−=4 2
3
m n
m n
− −
−= �
4 ( 3) 2 1m n− − − − −= 4 3 3m n− + −= 1 3m n− −=
3
1
mn=
Slide 5.1- 25
CLASSROOM EXAMPLE 8
Using the Definitions and Rules for Exponents
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
2 1
3
2 4y y
x x
−
2 2 1 1
6 1
2 4y y
x x
− −
−= �
2 1 1
5
2 4 y
x
−
=
2
5
2
4
y
x=
5
y
x=
Combination of rules
Slide 5.1- 26
CLASSROOM EXAMPLE 8
Using the Definitions and Rules for Exponents (cont ’d)
Simplify. Assume that all variables represent nonzero real numbers.
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Use the rules for exponents with scientific notation.
Objective 6
Slide 5.1- 27
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
In scientific notation, a number is written with th e decimal point after the first nonzero digit and multiplied by a p ower of 10.
This is often a simpler way to express very large or very small numbers.
Use the rules for exponents with scientific notatio n.
Slide 5.1- 28
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Scientific Notation
A number is written in scientific notation when it is expressed in the form
a × 10n
where 1 ≤ |a| < 10 and n is an integer.
Slide 5.1- 29
Use the rules for exponents with scientific notatio n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Converting to Scientific Notation
Step 1 Position the decimal point. Place a caret, ^, to the right of the first nonzero digit, where the decimal point will be placed.
Step 2 Determine the numeral for the exponent. Count the number of digits from the decimal point to the caret. This number gives the absolute value of the exponent on 10.
Step 3 Determine the sign for the exponent. Decide whether multiplying by 10n should make the result of Step 1 greater or less. The exponent should be positive to make the result greater; it should be negative to make the result less.
Slide 5.1- 30
Use the rules for exponents with scientific notatio n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write the number in scientific notation.
Step 1 Place a caret to the right of the 2 (the first nonzero digit) to mark the new location of the decimal point.
Step 2 Count from the decimal point 7 places, which is understood to be after the caret.
Step 3 Since 2.98 is to be made greater, the exponent on 10 is positive.
29,800,000
29,800,000 = 2.9,800,000. Decimal point moves 7 places to the left
29,800,000 = 2.98 × 107
Slide 5.1- 31
CLASSROOM EXAMPLE 9
Writing Numbers in Scientific Notation
Solution:^
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Step 3 Since 5.03 is to be made less, the exponent 10 is negative.
0.0000000503
0.0000000503 = 0.00000005.03
0.0000000503 = 5.03 × 10−8
Step 1 Place a caret to the right of the 5 (the first nonzero digit) to mark the new location of the decimal point.
Step 2 Count from the decimal point 8 places, which is understood to be after the caret.
Writing Numbers in Scientific Notation (cont’d)
Slide 5.1- 32
CLASSROOM EXAMPLE 9
Write the number in scientific notation.
^Solution:
Decimal point moves 7 places to the left
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Converting a Positive Number from Scientific Notation
Multiplying a positive number by a positive power of 10 makes the number greater, so move the decimal point to the right if n is positive in 10n.
Multiplying a positive number by a negative power of 10 makes the number less, so move the decimal point to the left if n is negative.
If n is 0, leave the decimal point where it is.
Slide 5.1- 33
Use the rules for exponents with scientific notatio n.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Write each number in standard notation.
2.51 ×103
–6.8 ×10−4
= 2510
= –0.000068
Move the decimal 3 places to the right.
Move the decimal 4 places to the left.
Slide 5.1- 34
CLASSROOM EXAMPLE 10
Converting from Scientific Notation to Standard Not ation
Solution:
When converting from scientific notation to standard notation, use the exponent to determine the number of places and the direction in which to move the decimal point.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Evaluate 200,000 0.0003.
0.06
×
5 4
2
2 10 3 10
6 10
−
−
× × ×=×
5 4
2
2 3 10 10
6 10
−
−
× × ×=×
1
2
2 3 10
6 10−
× ×=×
32 310
6
×= × 31 10= × 1000=
Slide 5.1- 35
CLASSROOM EXAMPLE 11
Using Scientific Notation in Computation
Solution:
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
The distance to the sun is 9.3 × 107 mi. How long would it take a rocket traveling at 3.2 × 103 mph to reach the sun?
d = rt, sod
tr
=
7
3
9.3 10
3.2 10
×=×
7 39.310
3.2−= ×
42.9 10≈ ×
It would take approximately 2.9 ×104 hours.
Slide 5.1- 36
CLASSROOM EXAMPLE 12
Using Scientific Notation to Solve Problems
Solution: