1 copyright © 2012, 2008, 2004 pearson education, inc. objectives 2 6 5 3 4 integer exponents and...

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- 2



Objective 1

Slide 5.1- 3


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- 4


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)

Slide 5.1- 5

CLASSROOM EXAMPLE 1

Using the Product Rule for Exponents

Solution:



Objective 2

Slide 5.1- 6


Zero ExponentIf a is any nonzero real number, then

a0 = 1.

Slide 5.1- 7


The expression 00 is undefined.


Evaluate.

290

(–29)0

–290

80 – 150

Slide 5.1- 8

CLASSROOM EXAMPLE 2

Using 0 as an Exponent


Negative ExponentFor any natural number n and any nonzero

real number a,

1 .nn

aa

A negative exponent does not indicate a negative number; negative exponents lead to reciprocals.

22

1 13 Not negative

3 9

22

1 13 Negati

9v

3e

Slide 5.1- 9



Write with only positive exponents.

6-5

(2x)-4

–7p-4

Evaluate: 4-1 – 2-1

Slide 5.1- 10

CLASSROOM EXAMPLE 3

Using Negative Exponents


Evaluate.

3

1

4

3

1

3

9

Slide 5.1- 11

CLASSROOM EXAMPLE 4

Using Negative Exponents


Special Rules for Negative Exponents

If a ≠ 0 and b ≠ 0 , then

and1 nna

a

.n m

m n

a b

b a

Slide 5.1- 12




Objective 3

Slide 5.1- 13


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- 14


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

3

5, 0

xy

y

Slide 5.1- 15

CLASSROOM EXAMPLE 5

Using the Quotient Rule for Exponents



Objective 4

Slide 5.1- 16


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- 17



Simplify, using the power rules.

45r

253y

33

4

Slide 5.1- 18

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 b

b a

Slide 5.1- 19



Write with only positive exponents and then evaluate.

42

3

Slide 5.1- 20

CLASSROOM EXAMPLE 7

Using Negative Exponents with Fractions

Solution:

51

, 02

xx


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- 21



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- 22



Simplify exponential expressions.

Objective 5

Slide 5.1- 23


Simplify. Assume that all variables represent nonzero real numbers.

524

4 6 8x x x

22

3

m n

m n

Slide 5.1- 24

CLASSROOM EXAMPLE 8

Using the Definitions and Rules for Exponents

Solution:


2 1

3

2 4y y

x x

Slide 5.1- 25

CLASSROOM EXAMPLE 8

Using the Definitions and Rules for Exponents (cont’d)

Simplify. Assume that all variables represent nonzero real numbers.



Objective 6

Slide 5.1- 26


In scientific notation, a number is written with the decimal point after the first nonzero digit and multiplied by a power of 10.

This is often a simpler way to express very large or very small numbers.


Slide 5.1- 27


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- 28



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- 29



Write the number in scientific notation.

29,800,000

Slide 5.1- 30

CLASSROOM EXAMPLE 9

Writing Numbers in Scientific Notation

346,100,000,000

102,000,000,000,000


0.0000000503

Writing Numbers in Scientific Notation (cont’d)

Slide 5.1- 31

CLASSROOM EXAMPLE 9

Write the number in scientific notation.

0.000000000385

0.00000000004088


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- 32



Write each number in standard notation.

2.51 ×103

–6.8 ×10−4

Slide 5.1- 33

CLASSROOM EXAMPLE 10

Converting from Scientific Notation to Standard Notation

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.

5.07 ×10−7


Evaluate 200,000 0.0003.

0.06

Slide 5.1- 34


Using Scientific Notation in Computation


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?

Slide 5.1- 35


Using Scientific Notation to Solve Problems

1 copyright © 2012, 2008, 2004 pearson education, inc. objectives 2 6 5 3 4 integer exponents and...

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