Transcript Chapter 8

Chapter 8
Exponents and
Exponential
Functions
Alexander Townsend
What Are Exponents
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A symbol or number placed above and after another symbol or
number to denote the power to which the latter is to be raised,
makes now sense right, here like this…
How it works is, take the larger
number on the left and multiply
it by itself the number of times
represented by the number on
the right. Exponents can be
written like you see … here →
Or by using a carrot symbol
2^2, 2^4, 2^6, etc.
Simple examples:
2^2=2x2 →
2x2=4
2^3=2x2x2 →
2x2x2=8
2^4=2x2x2x2→
2x2x2x2=16
Chapter 8 Section 1
Multiplication
Properties of Exponents
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Product of Powers: to multiply exponents that
have the same base number (number on the left)
you add there
powers (number on the right),
5^3 x 5^4 = 5^7 → 5x5x5x5x5x5x5 = 78125
watch it’s easy…
Also can be written like this… then you
multiply there powers
(3^4)^2 = 3^8 → 3x3x3x3x3x3x3x3 = 6561
• We can do the same thing even if we have different base numbers…
(4 x 5)^2 = 4^2 x 5^2 → (4x4) x (5x5) = 16x25= 400
•You can also do the same thing with variables…
(y^4)x(y^5)=(y^9) or (y^2z)^2= (y^4)(z^2)
As you know, exponents can be
positive numbers, negative
numbers, or zero.
Chapter 8 Section 2
Zero and Negative Exponents
Any number to the 0 power equals 1, looks like this…
2^0=1, 3^0=1 → There all the same, they all equal
4^0=1, 5^0=1
Zero as long as the base doesn't
Any negative exponent is the same as a positive exponent just with a 1 over it, you could call
a^0=1, y^0=1
equal zero… 0^0 ≠ 0
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negative and positive exponents reciprocals of each other…
3^(-2) = (1/3^2) So in words: Take the base and make it a fraction by making the numerator one, then
take away the negative exponent and make it positive
3^(-6)=(1/3^6)… 1/729
*Be careful negative exponents and negative bases are two different
things, Shown above is the way to evaluate negative exponents, here is
the way to solve negative bases,,, you know that (-)x(-)=(+), (+)x(+)=(+),
and that (+)x(-)=(-) it works the same way with exponents with negative
bases... (-2)^2=4 but (-2)^3=-8, if odd power answer is negative, even
power means positive answer.
Chapter 8 Section 3 Graphs of
Exponential Functions
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A simple exponential function is represented by y=ab^x, just
graph like any other graph, take your x coordinates make a
table and find y coordinates and plot points to form graph.
X
-2
-1
0
1
2
3
Y=
3(1/2)^x
12
6
3
3/2
3/4
3/8
From the graph of exponential functions
you can find there domain and range.
This graph has a domain of x=all real #’s
and a range of y=all real #’s > 0
Chapter 8 Section 4 Division
Properties of Exponents
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You know that Multiplication and Division are opposites, so the
same is true about multiplication properties of exponents and
division properties of exponents, they are also opposites.
For example… (6^4)x(6^2)=6^6, Multiplication Powers Property (add powers)
(6^4)/(6^2)=6^2, Division Powers Property (subtract powers)
More different examples…
(2/3)^2= (2^2)/(3^2)= 4/9, *don’t forget to distribute the power to top and bottom.
(-3/y)^3=(-3^3)/(y^3)= -27/y^3
Try to Simplify Exponential Functions that look hard…
(2z^2y/3z) X (9zy^2/y^4)= (18z^3y^3)/(3zy^4) →
6z^2y^-1 → (6z^2/y)
More Examples: 3^9/3^5 = 3^4
X^5/X^? = X^2 → ?=3
(1/6)^4 = 1/? → ?= 1296
*They all use the same basic rules its just to do
different problems.
Chapter 8 Section 5
Scientific Notation
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A method of expressing number in a different way, *multiplied by 10 to the
appropriate power (exponent).
That makes no sense right? But really its pretty easy... Rather then trying to
explain and losing half of you… just watch and learn…
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Scientific Notation…
2.83 x 10^1
4.9 x 10^5
8 x 10^-1
1.23 x 10^-3
→
→
→
→
Regular Notation…
28.3
490,000
0.8
0.00123
Now that we’ve tried a couple lets try to explain it…
• When you see Scientific notation the first thing to do is look at the power over the 10(*when dealing in Scientific
notation your base will always be 10). The power will tell you which way to move the decimal, because that is all
we are doing in Scientific Notation, (Multiplying by 10^a certain number makes your original numbers decimal move
either right or left a certain number of times). Here is the best and easiest way to explain it…
•If the power of 10 is positive you move the decimal to the right the number of times in the power.
•If the power of 10 is negative you move the decimal to the left the number of times in the power.
Chapter 8 Section 6
Exponential Growth
Functions
* Basic
Exponential
Growth
Function
Graph
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Something grows exponential if is increase the same amount in every unit of time. In other words if
increases at the same rate all the time. These can be tricky but with practice you’ll get them no problem.
The function you use to solve them is…
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y=C(1+ r)^t
nothing but blah,blah,blah let’s clear it up…
C or P= the amount you have before growth occurs, C can be P when dealing with $
r= the growth rate
t= time (the number of times the growth happens)
And both C and r are positive (1+ r) is the Growth Factor
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The problems usually start as a word problem, all you do is find what you need and plug it into the function
and solve it is easy after awhile,
Example:
*Let y be the weight of the salmon during the first six weeks and let t be the number of days. The
initial weight of the salmon C is 0.06. The growth rate (r) is 10%(0.10).
Set it Up…
y = C(1+r)^t
* The only hard
y = 0.06(1+ 0.10)^42 (42 days = 6 weeks)
part is finding
then simplify
what you need
y = 0.06(1.1)^42
in the question,
y = 3.29
* They are all use the same formula, if you
can do one you can do them all.
but once you do
they are easy.
Chapter 8 Section 7
Exponential Decay Function
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These will be easy for you they are the same as Exponential Growth function except on little difference,
they get solved the same way and everything there just a small difference in the formula you use…
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y=C(1- r)^t
C or P= the amount you have before decay occurs, C can be P when dealing with $
r= the decay rate
t= time (the number of times the decay happens)
See the difference C and r are negative (1 - r) is the Decay Factor
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→
*See that is the only difference they are all the same except the negative…
Example:
Let y be the value of the car and let t be the number of years of ownership. The initial value
of the car C is $ 16,000. The decay rate r is 12%, or 0.12.
y= C( 1 – r )^t
=16,000 ( 1- 0.12)^t
=16,000 (0.88)^t
Now take the decay over 8 years,
y= 16,000 (0.88)^8
= 5754
That’s a rap!
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Now you have the basic knowledge of
exponents. You can use them on into
geometry, Algebra 2, Pre Calc, Trig, and on
into Calculus. More rules and new concepts
will be added onto the hopefully strong
foundation set down for you in this class.