copyright © 2013, 2009, 2006 pearson education, inc. 1 section 5.7 negative exponents and...

Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

Section 5.7

NegativeExponents and

ScientificNotation



Objective #1 Use the negative exponent rule.


If b is any real number other than 0 and n is a natural number, then

When a negative number appears as an exponent, switch the position of the base (from numerator to denominator or denominator to numerator) and make the exponent positive. The sign of the base does not change.

1 1 and .n n

n nb b

b b

Negative Exponents in Numerators and Denominators


4

44

3

33

Write with positive exponents.

1

Write 5 with positive exponents.

15

5

x

xx

Negative Exponents

EXAMPLESEXAMPLES


4

44

3

33

Write with positive exponents.

1

Write 5 with positive exponents.

15

5

x

xx

Negative Exponents


Objective #1: Example

1a. Use the negative exponent rule to write 35 with a positive exponent and then simplify.

33

15

51

125

1b. Use the negative exponent rule to write 4( 3) with a positive exponent and then simplify.

44

1( 3)

( 3)

181



1a. Use the negative exponent rule to write 35 with a positive exponent and then simplify.

33

15

51

125

1b. Use the negative exponent rule to write 4( 3) with a positive exponent and then simplify.

44

1( 3)

( 3)

181



1c. Use the negative exponent rule to write 18 with a positive exponent and then simplify.

111

8818



1d. Write 24

5

with positive exponents only.

Then simplify, if possible.

2 2

24 55 4

2516



1d. Write 24

5



2 2

24 55 4

2516



1e. Write 2

1

7 y with positive exponents only.


2

21

77

y

y


1f. Write 1

8x

y



1 8

8 1

8

x y

y x

yx



Objective #2 Simplify exponential expressions.


Simplifying Exponential Expressions

An exponential expression is “simplified” when

a. Each base occurs only once.

b. No parentheses appear.

c. No powers are raised to powers.

d. No negative or zero exponents appear.



4444 22 baab

134919711971 QQQ

632

3

32

33

2

4913171717

xxxx

50105105 WWW

Simplification Techniques

Examples

If necessary, remove parentheses by using the Products to Powers Rule or the Quotient to Powers Rule.

If necessary, simplify powers to powers by using the Power Rule.



20164164 HHHH

51234523043 YX

6172317

23

VVV

V

1212

3131

KK

Simplification Techniques

Examples

Be sure each base appears only once in the final form by using the Product Rule or Quotient Rule

If necessary, rewrite exponential expressions with zero powers as 1. Furthermore, write the answer with positive exponents by using the Negative Exponent Rule


Simplify: 834

5

)2(

x

x

When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.2

2

8

6

8

323

5

85

8

5

8

5

)(2

x

x

x

x

x

x

Cube each factor in the numerator.

Multiply powers using (bm)n = bmn.

Division with the same base, subtract exponents.


EXAMPLEEXAMPLE


Simplify: 2 3

8

(2 )

5

x

x

When a negative number appears as an exponent, switch the position of the base. The x-2 moves from numerator to denominator as x2.

3 2 3

8

6

8

2

2

2 ( )

5

8

5

8

58

5

x

x

x

x

x

x

Cube each factor in the numerator.

Multiply powers using (bm)n = bmn.

Division with the same base, subtract exponents.


EXAMPLEEXAMPLE


2a. Simplify: 12 2x x

12 2 12 2

10

101

x x x

x

x




2b. Simplify: 3

975

5

x

x

3 3

9 9

3 9

6

6

75 7555

15

15

15

x x

x x

x

x

x



2c. Simplify: 4 2

11(6 )x

x

4 2 2 4 2

11 11

4 2

11

8

11

8 11

3

3

(6 ) 6 ( )

36

36

36

36

36

x x

x x

x

x

x

x

x

x

x



2d. Simplify:

58

4x

x

58 54

4

20

201

xx

x

x

x


Objective #3 Convert from scientific notation to

decimal notation.


Scientific Notation

At times you may find it necessary to work with really large numbers, or alternately, really small numbers. Here you will learn how to write these often cumbersome numbers in scientific notation.

Scientific Notation

A positive number is written in scientific notation when it is expressed in the form

Where a is a number greater than or equal to 1 and less than 10 and n is an integer.

10 ,na

(1 10)a


Scientific Notation

Converting from Scientific to Decimal Notation

We can use n, the exponent on the 10 in , to change a number in scientific notation to decimal notation. If n is positive, move the decimal point in a to the right n places. If n is negative, move the decimal point in a to the left places.

10na

n



3a. Write 97.4 10 in decimal notation. The exponent is positive so we move the decimal point nine places to the right.

97.4 10 7,400,000,000

3b. Write 63.017 10 in decimal notation. The exponent is negative so we move the decimal point six places to the left.

63.017 10 0.000003017


Objective #4 Convert from decimal notation to

scientific notation.


Converting from Decimal to Scientific Notation

• Determine a, the numerical factor. Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10.

• Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal was moved. The exponent n is positive if the given number is great than 10 and negative if the given number is between 0 and 1.

Scientific Notation



4a. Write 7,410,000,000 in scientific notation.

97,410,000,000 7.41 10

4b. Write 0.000000092 in scientific notation.

80.000000092 9.2 10


Objective #5 Compute with scientific notation.


Computations with Scientific Notation



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Perform the indicated computation, writing the answer in scientific notation.

86 102.31041.5

13107312.1

6 85.41 3.2 10 10

86 102.31041.5

Regroup factors

Multiply

Simplify

Rewrite in scientific notation

1410312.17 114107312.1

8610312.17



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Perform the indicated computation, writing the answer in scientific notation.

6

8

1041.5

102.3

Group factors

Divide

Simplify

Write in scientific notation

681059.0 21059.0

3109.5

6

8

10

10

41.5

2.3

6

8

1041.5

102.3



5a. Perform the indicated computation, writing the answer in

scientific notation: 8 2(3 10 )(2 10 )

8 2 8 2

8 2

10

(3 10 )(2 10 ) (3 2) (10 10 )

6 10

6 10



5b. Perform the indicated computation, writing the answer in

scientific notation: 7

48.4 10

4 10

7 7

4 4

7 ( 4)

11

8.4 10 8.4 1044 10 10

2.1 10

2.1 10


5c. Perform the indicated computation, writing the answer in

scientific notation: 2 3(4 10 )

2 3 3 2 3

6

5

(4 10 ) 4 (10 )

64 10

6.4 10



Objective #6 Solve applied problems using scientific notation.


Scientific Notation

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

The area of Alaska is approximately acres. The state was purchased in 1867 from Russia for $7.2 million. What price per acre, to the nearest cent, did the United States pay Russia?

6102.7

We will first write 7.2 million in scientific notation. It is:

81066.3

Now we can answer the question.


Scientific Notation

Therefore, since dollars/acre equals $0.0197/acre, then the price per acre for Alaska was approximately 2 cents per acre.

CONTINUEDCONTINUED

Divide ‘dollar amount’ by ‘acreage’

Divide

Simplify8

6

10

10

66.3

2.7

6

8

7.2 10

3.66 10

21097.1

861097.1 Subtract



6. The cost of President Obama’s 2009 economic stimulus package was $787 billion, or 117.87 10 dollars. If this cost were evenly divided among every individual in the United States (approximately 83.07 10 people), how much would each citizen have to pay?

11 11

38 8

7.87 10 7.87 102.56 10 2560

3.073.07 10 10

Each citizen would have to pay about $2560.

copyright © 2013, 2009, 2006 pearson education, inc. 1 section 5.7 negative exponents and...

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