Laws of Exponents. Exponential Notation Base Exponent Base raised to an exponent.
8.1 Exponents axax exponent base means a a a … a x times.
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Transcript of 8.1 Exponents axax exponent base means a a a … a x times.
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8.1 Exponents
ax
exponent
base
means a • a • a • …• a
x times
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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
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8.1 Exponents
42 ·45 47
32 ·33 35
(x2)5 x10
(x2)(x9) x11
(a2b3)7 a14b21
(3a3b5)4 81a12b20
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•For any nonzero number x:
nn
xx
1
andn
nx
x
1
8.2 Negative Exponents
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•For any nonzero number x:
10 x
8.2 Negative Exponents
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•Examples:
4
1
2
12
22
2733
1 33
4
9
2
3
3
222
8.2 Negative Exponents
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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
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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
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8.3 Division Property of Exponents
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
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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
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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
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8.4 Scientific Notation
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.
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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
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8.4 Scientific Notation
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
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8.4 Scientific Notation
Express in scientific notation
1) 421.96
2) 0.0421
3) 0.000 56
4) 467 000 000
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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
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Scientific Notation
7,000,000,000
= 7 billion
= 7 x 109
7,000,000
= 7 million
= 7 x 106
8.4 Scientific Notation
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Scientific Notation
7,240,000
= 7.24 million
= 7.24 x 106
.00345
= 345 ten thousandths
= 3.45 x 10-3
8.4 Scientific Notation
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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
8.4 Scientific Notation
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Multiplying
Add Exponents and Scientific Notation
(3.1 x 106)(2.0 x 102)
= 6.2 x 108
8.4 Scientific Notation
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DividingSubtract Exponents and Scientific Notation
(3.8 x 106)
(2.0 x 102)= 1.9 x 104
8.4 Scientific Notation
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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
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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
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8.4 Problem Solving
How old are you in seconds?
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DividingSubtract Exponents and Scientific Notation
(3.8 x 106)
(2.0 x 102)= 1.9 x 104
8.4 Scientific Notation
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DividingSubtract Exponents and Scientific Notation
(3.8 x 106)
(2.0 x 102)= 1.9 x 104
8.4 Scientific Notation
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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
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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
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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
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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
8.7 Exponential Growth and Decay
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?
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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
8.7 Exponential Growth and Decay
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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.
8.7 Exponential Growth and Decay
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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.
8.7 Exponential Growth and Decay
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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.
8.7 Exponential Growth and Decay
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