Download Chapter Eight: Exponents and Exponential Functions

Chapter Eight: Exponents and Exponential Functions
Section One: Laws of Exponents: Multiplying Monomials
An exponent is a shorthand way of writing repeated multiplication
35  3  3  3  3  3  243
In the expression 35 , 3 is the base and 5 is the exponent or power. Anything raised to the
first power is simply that number (Ex. 51  5 ).
EX1: Evaluate the following exponents.
a. 2 6
b. 43
c. 7 2
d. 81
There are many rules that we will be discussing in the remainder of the chapter. The first
rule is the Product-of-Powers Property:
When the expressions are being multiplied and the bases are the same then we can add
powers. This is why:
32  35   3  3   3  3  3  3  3  37
x3  x 6   x  x  x    x  x  x  x  x  x   x 9
EX2: Simplify.
a. 32  33
b. j 3  j 4
c. 6 2  6
d. 7 a  7b
EX3: Find the missing power.
a. 56  5 x  513
b. z 2  z x  z 7
A monomial is a constant (number), a variable, or the product of a number and one or
more variables. In the last case the number is called a coefficient. When multiplying
multiple monomials we can remove parenthesis, regroup using the commutative property,
and then simplify using multiplication or Product-of-Powers.
EX4: Simplify.
a.  2c 2  3c 3 
b.  3xy 2  5 x 2 y  y 3 z 
c.  3m 2  60mp 2 
d. x 3c  x 4 c
Section Two: Laws of Exponents: Powers of Products
The next rule that we will discuss is Powers-of-Powers Property. We can use the Productof-Powers Property to show that it is true.
 2    2  2  2  2   2
3 4
3
3
3
3
 x    x  x  x   x
5 3
12
5
5
5
15
When a power is raised to another power, we can multiply the powers.
EX1: Simplify.
a.  26 
2
b. 104 
c.  p 2 
3
5
d.  x m 
2
The Power-of-a-Product Property is similar to the previous property.
 x y    x y  x y  x y  x y   x
3
2 4
3
2
3
2
3
2
3
2
12
y8
When a product is raised to a power, we can distribute the power to each factor.
EX2: Simplify.
a.  xy 3 
2
b.  x 2 y 2 z 2 
3
c.  x m y 2 z 
d.  abcd 
4
n
When we raise negative variables to powers we need to treat it like a 1 .
EX3: Simplify.
4
a.   y 
b.  y 4
c.  3y 
d. 3y 4
4
Section Three: Laws of Exponents: Dividing Monomials
When we are dividing monomials, we think about canceling factors.
x 7 x  x  x  x  x  x  x x  x  x  x  x  x  x x  x  x



 x3
4
x
xxxx
x  x  x  x
1
z2
zz
z  z
1
1




z 5 z  z  z  z  z z  z  z  z  z z  z  z z 3
We can simply subtract to see how many will be left after canceling.
EX1: Simplify.
1012
a.
104
210
b. 5
2
y10
c.
y
10 y1
d.
10
We can simplify division of monomials by using the same property.
EX2: Simplify.
3x3 y 4
a.
2 x 2 y 2
ab 2c 3
b.
ac 2
5 y 2  81t 4 

c.
45t 3
EX3: The volume, V, of a cube can be found with the formula V  e3 , where e represents
the length of one edge of the cube. Its surface area, S, can be found with the formula
S  6e 2 . Find the ratio of the volume of a cube to its surface area.
The Power-of-Fractions Property works pretty much the same as the Power-of-Products
Property.
4
4
 2   2  2  2  2  2
         4
 3   3  3  3  3  3
EX3: Simplify.
2
a.  
3
4
6
b.  
2
3
 c2 
c.  3 
d 
4
Section Four: Negative and Zero Exponents
In order to work with negative exponents we must first make them positive. To do so all
we need is a fraction bar. A factor with a negative exponent when moved across a
fraction bar results in a positive exponent.
EX1: Simplify.
a. 2 3
b. 32
1
c. 3
5
To simplify some expressions seen in the lesson, we can use the quotient of powers rule.
EX2: Simplify.
a. 35  38
105
b. 7
10
c. 35  37
52
d. 2
5
Anything raised to the zero power is always one. Also, any time that everything cancels
when using the quotient property we put a 1.
EX3: Simplify.
b6  b 2
a.
b4
b. 12x 6 y 21 
0
c. ((((34 )2 )5 )0 )6
EX4: Simplify the expressions.
a. 9a 2b3  2a5b3 
 m9n 
b.  10 6 
 5m n 
3
2
Section Five: Scientific Notation
Since many times scientists deal with really large and really small numbers, they have a
special system of writing these numbers. There are two rules for writing numbers in
scientific notation:
1. It must be in the form: a 10n
2. a must be a number between 1 and 10
EX1: Write the following numbers in scientific notation.
a. 875,000
b. 3,700,000,000
c. 0.0000000402
d. 0.000000000106
EX2: Write each number in standard notation.
a. 6.725 105
b. 3.677 1011
c. 3.02 104
d. 4.36 107
We can use our rules of exponents to simplify scientific notation expressions.
EX3: Simplify.
a. 3 1028  104
b.  2.5 105 1.56 103 
c.
3.6 105
6 103
EX4: The speed of light is about 1.86 105 miles per second. When Pluto is closest to the
sun, the distance from the sun to Pluto is approximately 2.76 109 miles. How long does
it take sunlight to reach Pluto?
Section Six: Exponential Functions
We studied about linear functions back in the five. In this section we will discuss a new
type of function called an exponential function. Things such as bacteria grow
exponentially. We can see an example below of a bacteria growth that doubles each hour.
Time
0 1 2 3
4
5
6
7
8
Bacteria 1 2 4 8 16 32 64 128 256
The basic form of an exponential equation:
A  P 1  r 
t
where A is the final amount, P is the starting amount, r is the rate of growth, and t is the
time it grows.
EX1: When Chris was born, his parents put $1000 into his savings account. The amount
of money in the account is guaranteed to increase at a yearly rate of 4.5%. Assume that
no additional deposits are made to the account and no withdrawals are made. By what
amount will the account increase in 18 years?
EX2: Make a table of values and graph the functions
a. y  2 x
b. y  3x
1
c. y   
2
x
Section Seven: Applications of Exponential Functions