Chapter 5 Section 1 - Houston Community College

Section 1Chapter 5

1

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


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



Objective 1

Slide 5.1- 4


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


Be careful not to multiply the bases. Keep the same base and add the exponents.


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:



Objective 2

Slide 5.1- 7


Zero ExponentIf a is any nonzero real number, then

a0 = 1.

Slide 5.1- 8


The expression 0 0 is undefined.


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:


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



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:


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:


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




Objective 3

Slide 5.1- 14


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


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.


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:



Objective 4

Slide 5.1- 17


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



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:


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



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=


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



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



Simplify exponential expressions.

Objective 5

Slide 5.1- 24


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:


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:


Use the rules for exponents with scientific notation.

Objective 6

Slide 5.1- 27


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


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



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



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:^


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


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



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.


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


Using Scientific Notation in Computation

Solution:


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


Using Scientific Notation to Solve Problems

Solution:

Chapter 5 Section 1 - Houston Community College

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