Math 72 Section 5.2

• Class Participation: – Exercise 5.2, #80 Use the formula A = P(1 +r)t to find the value of an $8,500

investment compounded at 6.5% at the end of a ten year period.

A = P(1 +r)t A = 8500(1 +.065)10 A = 8500(1.065)10

A = $15,955.67

Math 72 Section 5.2

• Class Participation: – Exercise 5.2, #82

Use a calculator to complete the given table of values. Then determine whether Y1 = Y2 or Y1 = Y3

x Y1 Y2 Y3

–2 –1 0 1 2

Y1 =x 3

x12

Y2 =1x 9

Y3 =1x 4

x Y1 Y2 Y3

–2 –.002 –.002 .0625 –1 –1 –1 1 0 undefined undefined undefined

1 1 1 1 2 .00195 .00195 .0625

Y1 = Y2

Quiz 19

• Get out paper and pencil or pen – 8.5x11 sheet of paper; fold vertically • Put your name outside at top

• Put notes away • You will have 5 minutes to complete the

problem

Quiz 19

Simplify. Assume bases nonzero:

Show your work.

5x 3y 5( )2

3xy 3( )4 =

25x 6y10

81x 4 y12 =25x 2

81y 2

Here is the solution:

Math 72 Section 5.3

Negative Exponents

Use All Properties of Exponents

Scientific Notation

Example Problem

• Each song on a music player requires about 4x106 bytes of memory. If the player has 80 GB (8 x 1010) bytes available, approximately how many songs will it hold?

Negative Exponents a. We stated the quotient rule as for x ≠ 0 and n > m.

Simplify

b. Assume the quotient rule is true for all integers m and n.

Simplify

c. Since we want the two expressions to be the same we must

have:

xm

xn =1

xn −m

x 4

x 7 =

1x 7−4 =

1x 3

xm

xn = xm −n

x 4

x 7 =

x 4 −7 = x −3

x −3 =1x 3

1.

Negative Exponents

Algebraic Verbal Example For any nonzero real number x and natural

number n,

x −n =1xn

A _______ base with a negative exponent can be rewritten by

using the ________ of the base and the corresponding

positive exponent.

nonzero

reciprocal

x −4 =1x 4

Negative Exponents

2−4 =

124 =

116

−24 =

−(24 ) = −16

3−1 + 5−1 =1 13 5

+ =

(3+ 5)−1 =

(8)−1 =18

2. 3.

4.

5.

5 1 3 15 3 3 5

⋅ + ⋅ =5 3 8

15 15 15+ =

Negative Exponents

• Note the effect of a negative exponent on a fraction:

Simplify:

23

−1

=

123

=1 ÷23

=1⋅32

=32

6.

Fraction to Negative Exponent

Algebraic Verbal Example For any nonzero real numbers x and y, and

natural number n,

xy

−n

=yx

n

A nonzero fraction to a negative exponent

can be written by taking the ________ of the fraction and

using the corresponding

positive exponent.

reciprocal

27

−2

=72

2

=494

Negative Exponents

25

−3

=

52

3

=125

8

23

−2

=

32

2

=94

13−2 =

32

1= 9

2−2

3= 2

1 1 13 2 3 4 12

= =⋅ ⋅

7. 8.

9. 10.

Negative Exponents

5x( )−2 =

15x( )2 =

125x 2

5x −2 =

5x 2

3x −2

y=

3x 2y

−3xy

−2

=

y−3x

2

=y 2

9x 2

11. 12.

13. 14.

Summary of Exponent Rules

Product rule:

Power rules:

Quotient rule:

Zero exponent:

Negative exponent rule:

xm ⋅ xn =

xm +n

xm( )n=

xm⋅n

xy( )m=

xm ym

xy

m

=

xm

ym

xm

xn =

xm −n

x 0 =

1, x ≠ 0

x −n =

1xn

Exponents Simplify to form with positive exponents. x ≠ 0, y ≠ 0

x 5

x −3

4

= ( )45 3x x =

( )( ) 22 53 2x x−

− = ( ) 236x−

=

15.

16.

( )48 32x x=

( )2 63

1 1366 xx

=

Exponents Simplify to form with positive exponents. x ≠ 0, y ≠ 0

x 2y −3

x −1y 4 =

x 2x1

y 4 y 3 =

5x −3y −5( )2

3xy−3( )4 =

25x −6y −10

81x 4 y −12 =

17.

18.

x 3

y 7

25y12

81x 4 x 6y10 =

25y 2

81x10

Exponents Simplify to form with positive exponents. x ≠ 0, y ≠ 0

36x 3

12x −2

15x −7

45x 4

=

3x 5

3x11 =1x 6

14x −3y 2

35x 2y −4

−2

=

2y 2y 4

5x 2x 3

−2

=

3x 3x 2( ) 13x 4 x 7

=

2y 6

5x 5

−2

=

5x 5

2y 6

2

=25x10

4y12

19.

20.

Exponents Simplify to form with positive exponents. x ≠ 0, y ≠ 0

−2x −3y 4( )24x 3y −6( )−1

=

−2x −3y 4( )2

4x 3y −6( )1 =

10x −2y 4( )−2−2x 6y −1( )

25xy−4( )−1 =

25xy−4( ) −2x 6y −1( )10x −2y 4( )2 =

4x −6y 8

4x 3y −6 =

y 8y 6

x 3x 6 =y14

x 9

−50x 7y −5

100x −4 y 8 =

−x 7x 4

2y 8y 5 = −x11

2y13

21.

22.

Scientific Notation

Examples:

7.498x1012 = 7498000000000

2.4933x102 = 249.33

6.3455x10–5 = 0.000063455

5.001x10–22 = 0.0000000000000000000005001

Scientific Notation

Writing a number in standard decimal notation: Verbal Example a. If the exponent on 10 is positive, move decimal point to the ____. right

c. If the exponent on 10 is negative, move decimal point to the ___. left

b. If the exponent on 10 is ____, do not move the decimal point.

zero

a. 3.456x102 = 3.456x100 = 345.6 The decimal is moved 2 places to the right.

b. 3.456x100 = 3.456x1 = 3.456 The decimal is not moved.

c. 3.456x10–2 = 3.456x0.01 = 0.03456. The decimal is moved 2 places to the left.

Scientific Notation • Write in standard decimal notation:

5.71x104 = 4.25x10–4 = 3.2x10–6 = 3.987x107 =

57,100 0.000425

0.0000032 39,870,000

23.

22. 26.

24.

Scientific Notation Writing a number in scientific notation:

Verbal Example

b. The exponent on 10 is _______ if the original number is less than 1.

1. Move the decimal point immediately to the ____ of the first nonzero digit of the number.

right

a. The exponent on 10 is 0 or positive if the original number is 1 or ______.

negative

2. Multiply by a power of 10 determined by the number of places the decimal was moved.

greater

3.456 = 3.456x100

345.6 = 3.456x102

0.03456 = 3.456x10–2

Scientific Notation • Write in scientific notation:

80,000 = 72,300 = 0.008 =

0.0000985 =

8.0x104

7.23x104

8.0x10–3

9.85x10–5

27.

29. 30.

28.

Scientific Notation • On your calculator press MODE→ENTER2ndMODE then

(2.4(2nd)(comma)(-)8)(3.1(2nd)(comma)11) then enter • Your screen should show: • (2.4E–8)(3.1E11) • 7.44E3 • Write this in scientific notation and standard decimal notation

0.00000002.4 x 310,000,000,000 = 7440

2.4 x 10–8 x 3.1 x 1011 = 7.44 x 103

31.

Scientific Notation

Each song on a music player requires about 4x106 bytes of memory. If the player has 80 GB (8 x 1010) bytes available, approximately how many songs will it hold?

8.0 ×1010

4 ×106 = 2 ×104 = 20,000 songs

32.

Scientific Notation Use scientific notation to estimate:

(4,990,000)(0.000147) 4,990,000 ≈ 5 x 106

Calculator: (4990000)(0.000147) =

733.53

0.000147 ≈ 1.5 x 10–4

(5 x 106)(1.5 x 10–4) = 7.5 x 106–4 = 7.5 x 102 = 750

33.

Exponents Evaluate each expression for x =2 and y = –3:

−x −2 + y −2 =

−1x 2 +

1y 2 =

−12( )2 +

1−3( )2 =

−14

+19

=−936

+436

=

−536

34.

Exponents Evaluate each expression for x =2 and y = –3:

− x + y( )2=

− 2 + −3( )( )2=

−1

− −1( )2 =

35.

Exponents

Evaluate each expression for x =2 and y = –3:

x + y( )−2=

1x + y( )2 =

12 + −3( )( )2 =

1−1( )2 =1

35.

Math 72 Section 5.3

• Class Participation: – Exercise 5.3, #56