8.1 exponents axax exponent base means a a a … a x times

8.1 Exponents

ax

exponent

base

means a • a • a • …• a

x times

8.1 Exponents

Rules For Exponents and Scientific NotationIf a > 0 and b > 0, the following hold true for all real numbers x and y.

1. ax ay axy

2. ax

ayax y

3. ax yaxy

4. ax bx (ab)x

5. a

b

x

ax

bx

6. a0 1

7. a-x 1

ax

8. ap

q apq

8.1 Exponents

42 ·45 47

32 ·33 35

(x2)5 x10

(x2)(x9) x11

(a2b3)7 a14b21

(3a3b5)4 81a12b20

•For any nonzero number x:

nn

xx

1

andn

nx

x

1

8.2 Negative Exponents

•For any nonzero number x:

10 x


•Examples:

4

1

2

12

22

2733

1 33

4

9

2

3

3

222


5

3

x

x

xxxxx

xxx

53x

2

1

x

If we apply the quotient rule, we get:

5

3

x

x 2x 2

1

x

8.3 Division Property of Exponents


16

9

32

x

9

4 1

yy

5

1

y

2

4

3

2

43

7

3

5

2

xy

yx

x

xy

3

5

4

35

6 23 yx

64

125

3

8

x


125

8

3

3

4

x

y

2

342 1

xyyx

10

5

y

x

3

2

5

34

52

5

23

10

6

4

5

yx

yx

x

yx

3

2

1

4

4

4

3

x

y

8

9

364

x

y

Scientific notation is used to express very large or very small numbers.

A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10.

The number has one digit to the left of the decimal point.

The power of ten indicates how many places the decimal point was moved. .

8.4 Scientific Notation

8.4 Scientific Notation 5 500 000 = 5.5 x 106 We moved the decimal 6 We moved the decimal 6

places to the left.places to the left.

A number between 1 and 10A number between 1 and 10


0.0075 = 7.5 x 10-3

We moved the decimal 3 We moved the decimal 3 places to the rightplaces to the right.

A number between 1 and 10A number between 1 and 10

Numbers less than 1 Numbers less than 1 will have a negative will have a negative exponent.exponent.

8.4 Scientific Notation CHANGE SCIENTIFIC NOTATION TO

STANDARD FORM

2.35 2.35 xx 101088

= 2.35 = 2.35 xx 100 000 000 100 000 000

= 235 000 000= 235 000 000

Standard formStandard form

Move the decimal 8 places to the rightMove the decimal 8 places to the right


9 x 10-5

= 9 x 0.000 01

= 0.000 09

Move the decimal 5 places to the leftMove the decimal 5 places to the left

Standard formStandard form


Express in scientific notation

1) 421.96

2) 0.0421

3) 0.000 56

4) 467 000 000


Change to Standard Form

1) 4.21 x 105

2) 0.06 x 103

3) 5.73 x 10-4

4) 4.321 x 10-5

Scientific Notation

7,000,000,000

= 7 billion

= 7 x 109

7,000,000

= 7 million

= 7 x 106


Scientific Notation

7,240,000

= 7.24 million

= 7.24 x 106

.00345

= 345 ten thousandths

= 3.45 x 10-3


Adding and Subtracting

Exponents and Scientific Notation must be the same!

(1.2 x 106) + (2.3 x 105)

change to

(1.2 x 106) + (0.23 x 106)

= 1.43 x 106


Multiplying

Add Exponents and Scientific Notation

(3.1 x 106)(2.0 x 102)

= 6.2 x 108


DividingSubtract Exponents and Scientific Notation

(3.8 x 106)

(2.0 x 102)= 1.9 x 104


8.4 Problem Solving

The distance from the earth to the sun is

1.5 x 1011 m The speed of light is 3 x 108 m/s.

How long does it take for light from the sun to reach the earth?

smx

mx

r

dt

/103

105.18

11

ssmx

mx

r

dt 500

/103

10158

10

8.4 Problem Solving

The mass of an electron is 9.11 x 10-31 kg and the mass of a proton is 1.67 x 10-27 kg. How many times bigger is the proton than the electron?

kgxnkgx 2731 1067.1)(1011.9

18001011.9

1067.131

27

kgx

kgxn

8.4 Problem Solving

How old are you in seconds?

DividingSubtract Exponents and Scientific Notation

(3.8 x 106)

(2.0 x 102)= 1.9 x 104


If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the equation

y C r t ( )1Where

C = initial amount

r = growth rate (percent written as a decimal)

t = time where t > 0

(1+r) = growth factor where 1 + r > 1

8.6 Compound Interest and Exponential Growth

You deposit $1500 in an account that pays 2.3% interest compounded yearly,

1) What was the initial principal (P) invested?

2) What is the growth rate (r)? The growth factor?

3) Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money?

3 ) A P r

A

A

t

( )

( . )

$ .

1

1500 1 0 023

1569 79

2

1) The initial principal (P) is $1500.

2) The growth rate (r) is 0.023.

The growth factor is 1.023.

8.6 Compound Interest and Exponential Growth

If a quantity decreases by the same proportion r in each unit of time, then the quantity displays exponential decay and can be modeled by the equation

y C r t ( )1Where

C = initial amount

r = growth rate (percent written as a decimal)

t = time where t > 0

(1 - r) = decay factor where 1 - r < 1

8.7 Exponential Growth and Decay

1) The initial investment was $22,500.

2) The decay rate is 0.07. The decay factor is 0.93.

3 1

22 500 1 0 07

20 925

1

) ( )

, ( . )

$ ,

y C r

y

y

t

y C r

y

y

t

( )

, ( . )

$ .

1

22 500 1 0 07

19460 25

2


You buy a new car for $22,500. The car depreciates at the rate of 7% per year,

1. What was the initial amount invested?

2. What is the decay rate? The decay factor?

3. What will the car be worth after the first year? The second year?

1) Make a table of values for the function using x-values of –2, -1, 0, 1, and 2. Graph the function. Does this function represent exponential growth or exponential decay?

yx

1

6


x yx

1

6

2 1

66

22

1 1

66

11

0 1

6

0

1 1

6

1

2 1

6

2

y

36

6

11

61

36

This function represents exponential decay.


C = $25,000

T = 12

R = 0.12

Growth factor = 1.12

y C r

y

y

y

t

( )

$ , ( . )

$ , ( . )

$ , .

1

25 000 1 0 12

25 000 1 12

97 399 40

12

12

Your business had a profit of $25,000 in 1998. If the profit increased by 12% each year, what would your expected profit be in the year 2010? Identify C, t, r, and the growth factor. Write down the equation you would use and solve.


Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Identify C, t, r, and the decay factor. Write down the equation you would use and

solve.


C = 25 mg

T = 4

R = 0.5

Decay factor = 0.5

y C r

y m g

y m g

y m g

t

( )

( . )

( . )

.

1

25 1 0 5

25 0 5

1 56

4

4

8.1 exponents axax exponent base means a a a … a x times

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