8.1 Exponents

1
8.1 Exponents
ax
2
8.1 Exponents
Rules For Exponents and Scientific Notation If a
gt 0 and b gt 0, the following hold true for all
real numbers x and y.
3
8.1 Exponents
47
42 45
a14b21
(a2b3)7
32 33
35
(3a3b5)4
81a12b20
(x2)5
x10
(x2)(x9)
x11
4
8.2 Negative Exponents

• For any nonzero number x

and
5
8.2 Negative Exponents

• For any nonzero number x

6
8.2 Negative Exponents

• Examples

7
8.3 Division Property of Exponents
If we apply the quotient rule, we get
8
8.3 Division Property of Exponents
9
8.3 Division Property of Exponents
10
8.4 Scientific Notation
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. .

11
8.4 Scientific Notation

• 5 500 000
• 5.5 x 106

We moved the decimal 6 places to the left.
A number between 1 and 10
12
8.4 Scientific Notation
Numbers less than 1 will have a negative exponent.

• 0.0075
• 7.5 x 10-3

We moved the decimal 3 places to the right.
A number between 1 and 10
13
8.4 Scientific Notation

• CHANGE SCIENTIFIC NOTATION TO STANDARD FORM

2.35 x 108
2.35 x 100 000 000 235 000 000
Standard form
Move the decimal 8 places to the right
14
8.4 Scientific Notation

• 9 x 10-5
• 9 x 0.000 01
• 0.000 09

Standard form
Move the decimal 5 places to the left
15
8.4 Scientific Notation

• Express in scientific notation
• 1) 421.96
• 2) 0.0421
• 3) 0.000 56
• 4) 467 000 000

16
8.4 Scientific Notation

• 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

17
Scientific Notation
8.4 Scientific Notation

• 7,000,000
• 7 million
• 7 x 106

• 7,000,000,000
• 7 billion
• 7 x 109

18
Scientific Notation
8.4 Scientific Notation

• .00345
• 345 ten thousandths
• 3.45 x 10-3

• 7,240,000
• 7.24 million
• 7.24 x 106

19
Adding and Subtracting
8.4 Scientific Notation

• 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

20
Multiplying
8.4 Scientific Notation

• Add Exponents and Scientific Notation
• (3.1 x 106)(2.0 x 102)
• 6.2 x 108

21
Dividing
8.4 Scientific Notation

• Subtract Exponents and Scientific Notation

1.9 x 104
22
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?

23
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?

24
8.4 Problem Solving

• How old are you in seconds?

25
Dividing
8.4 Scientific Notation

• Subtract Exponents and Scientific Notation

1.9 x 104
26
Dividing
8.4 Scientific Notation

• Subtract Exponents and Scientific Notation

1.9 x 104
27
8.6 Compound Interest and Exponential Growth
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

• Where
• C initial amount
• r growth rate (percent written as a decimal)
• t time where t gt 0
• (1r) growth factor where 1 r gt 1

28
8.6 Compound Interest and Exponential Growth

• You deposit 1500 in an account that pays 2.3
  interest compounded yearly,
• What was the initial principal (P) invested?
• What is the growth rate (r)? The growth factor?
• Using the equation A P(1r)t, how much money
  would you have after 2 years if you didnt
  deposit any more money?

• The initial principal (P) is 1500.
• The growth rate (r) is 0.023.
• The growth factor is 1.023.

29
8.7 Exponential Growth and Decay
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

• Where
• C initial amount
• r growth rate (percent written as a decimal)
• t time where t gt 0
• (1 - r) decay factor where 1 - r lt 1

30
8.7 Exponential Growth and Decay

• You buy a new car for 22,500. The car
  depreciates at the rate of 7 per year,
• What was the initial amount invested?
• What is the decay rate? The decay factor?
• What will the car be worth after the first year?
  The second year?

1. The initial investment was 22,500.
2. The decay rate is 0.07. The decay factor is 0.93.

31
8.7 Exponential Growth and Decay
32
8.7 Exponential Growth and Decay






This function represents exponential decay.
33
8.7 Exponential Growth and Decay
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.
C 25,000 T 12 R 0.12 Growth factor 1.12
34
8.7 Exponential Growth and Decay

• 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