exponents and polynomials

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Chapter 12

Exponents and Polynomials


12.2

Negative Exponents and Scientific Notation

Martin-Gay, Developmental Mathematics, 2e 33


Negative Exponents

Using the quotient rule,

0

2646

4

x

xxx

x

But what does x-2 mean?

26

4 11

xxxxxxxxx

xxxx

x

x



In order to extend the quotient rule to cases where the difference of the exponents would give us a negative number we define negative exponents as follows.

If a is a real number other than 0, and n is an integer, then

nn

aa

1

Negative Exponents



23

1

9

1

7

1

x

4

2

x

2a. 3

7b. x

4c. 2x

Example

Simplify by writing each result using positive exponents only.

Don’t forget that since there are no parentheses, x is the base for the exponent –4.

Don’t forget that since there are no parentheses, x is the base for the exponent –4.

Helpful HintHelpful Hint



Simplify by write each result using positive exponents only.

Example

2b. 3 23

1

9

1

2c. ( 3) 2)3(

1

9

1

31x

3a. x



Simplify by writing each of the following expressions with positive exponents.

3

1x

a.

3

11

x

b. 4

2

y

x

4

2

1

1

y

x2

4

x

y

1

3x 3x

(Note that to convert a power with a negative exponent to one with a positive exponent, you simply switch the power from the numerator to the denominator, or vice versa, and switch the exponent to its opposite value.)

Example



If m and n are integers and a and b are real numbers, then:

Product rule for exponents am · an = am+n

Power rule for exponents (am)n = amn

Power of a product (ab)n = an · bn

Power of a quotient , 0n n

n

a ac

c c

Quotient rule for exponents 0, aaa

a nmn

m

Zero exponent a0 = 1, a ≠ 0

Negative exponent 0,1

aa

an

n

Summary of Exponent Rules



Simplify by writing the following expression with positive exponents.

22 3 1

4 7 3

3

3

a b

a b

22 3 1

24 7 3

3

3

a b

a b

2 2 22 3

2 2 24 7 3

3

3

a b

a b

4 6 2

8 14 6

3

3

a b

a b

4 8 14 6 2 63 a b 4 8 43 a b 8

481

a

b

4 14 2

8 6 6

3

3

a b

a b

8

4 43

a

b

Example



In many fields of science we encounter very large or very small numbers. Scientific notation is a convenient shorthand for expressing these types of numbers.

A positive number is written in scientific notation if it is written as the product of a number a, where 1 ≤ a < 10, and an integer power r of 10: a ×10r.

Scientific Notation



To Write a Number in Scientific Notation

Step 1: Move the decimal point in the original number so that the new number has a value between 1 and 10.

Step 2: Count the number of decimal places the decimal point is moved in Step 1. If the original number is 10 or greater, the count is positive. If the original number is less than 1, the count is negative.

Step 3: Multiply the new number in Step 1 by 10 raised to an exponent equal to the count found in Step 2.

Scientific Notation



Write each of the following in scientific notation.

4700a. Move the decimal 3 places to the left, so that the new number has a value between 1 and 10.

Since we moved the decimal 3 places, and the original number was > 10, our count is positive 3.

4700 = 4.7 103

0.00047b. Move the decimal 4 places to the right, so that the new number has a value between 1 and 10.

Since we moved the decimal 4 places, and the original number was < 1, our count is negative 4.

0.00047 = 4.7 10-4

Example



In general, to write a scientific notation number in standard form, move the decimal point the same number of spaces as the exponent on 10. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

Scientific Notation



Write each of the following in standard notation.

5.2738 103a.

Since the exponent is a positive 3, we move the decimal 3 places to the right.

5.2738 103 = 5273.8

6.45 10-5b.

Since the exponent is a negative 5, we move the decimal 5 places to the left.

00006.45 10-5 = 0.0000645

Example



Operations with Scientific Notation

Example:

Multiplying and dividing with numbers written in scientific notation involves using properties of exponents.

Perform the following operations.

= (7.3 · 8.1) (10-2 · 105) = 59.13 103

= 59,130

(7.3 10-2)(8.1 105)a.

b. 9

4

104

102.1

9

4

10

10

4

2.1 5103.0 000003.0

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