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Section 5.1
Exponents and Scientific Notation
Exponents and Scientific Notation
• Exponents that are natural numbers are
shorthand notation for repeating factors.
• 34 = 3 • 3 • 3 • 3
• 3 is the base
• 4 is the exponent (also called power)
• Note, by the order of operations, exponents
are calculated before all other operations,
except expressions in parentheses or other
grouping symbols.
Product Rule (applies to common bases only)
am • an = am+n
Example
Simplify each of the following expressions.
32 • 34 = 32+4 = 36 = 3 • 3 • 3 • 3 • 3 • 3 = 729
x4 • x5 = x4+5 = x9
z3 • z2 • z5 = z3+2+5 = z10
(3y2)(-4y4) = 3 • y2 • -4 • y4 = (3 • -4)(y2 • y4) = -12y6
Zero exponent
a0 = 1, a  0
Note: 00 is undefined.
Example
(Assume all variables have nonzero values.)
Simplify each of the following expressions.
50 = 1
(xyz3)0 = x0 • y0 • (z3)0 = 1 • 1 • 1 = 1
-x0 = -1∙x0
= -1 ∙1 = -1
Problem from today’s homework:
Quotient Rule (applies to common bases only)
am
mn

a
an
a0
Example
Simplify the following expression.
4
7
9a b




9
a
b
 
3 5
41
72





3
a
b
    2   3(a )(b )
2
3ab
 3  a  b 
4 7
Group common
bases together
Problem from today’s homework:
Using the quotient rule,
4
x
46
2
x x
6
x
x0
But what does x -2 mean?
x
x x x x
1
1


 2
6
x
x x x x x x x x x
4
In order to extend the quotient rule to cases
where the difference of the exponents would
give us a negative number we define
negative exponents as follows:
If a  0, and n is an integer, then
a
n
1
 n
a
Example
Simplify by writing each of the following
expressions with positive exponents or
calculating.
1
1
2
 2 
1) 3
3
9
1
7
2) x  7
x
3)
2x
4
2
 4
x
Remember that without parentheses, only
the x is the base for the exponent –4, not
the entire expression 2x.
Note that in the previous problem, 2x-4
gave us 2 on top and x4 on the bottom,
and only the x term, not the 2,
was raised to the -4th power.
Here’s a question for you:
What would (2x)-4 look like when simplified?
Would it look different than 2x-4?
(Good question for a quiz!)
Example
Simplify by writing each of the following
expressions with positive exponents or
calculating.
1
3
1)  x  
3
x
1
1
2
2)  3   2  
3
9
1 Notice the difference in
1
2
3) ( 3) 

results when parentheses
2
9 enclose the -3
(3)
Example
Simplify by writing each of the following
expressions with positive exponents.
1)
2)
3
1
x
1
3

x
3 
1
x
1
x3
1
(Note that to convert a power with a negative
2
4
2
x
y exponent to one with a positive exponent, you
x

 2 simply switch the power from a numerator to
4
1
y
x a denominator, or vice versa, and switch the
exponent to its positive value.)
4
y
Problem from today’s homework:
Problem from today’s homework:
• In many fields of science we encounter very
•
large or very small numbers. Scientific
notation is a convenient shorthand for
expressing these types of numbers using
powers of the base 10.
A positive number is written in scientific
notation if it is written as a product of a
number a, where 1  a < 10, and an integer
power r of 10.
a  10r
To Write a Number in Scientific Notation
1) Move the decimal point in the original
number to the left or right, so that the new
number has a value between 1 and 10.
2) Count the number of decimal places the
decimal point is moved in Step 1.
•
•
If the original number is 10 or greater, the count is
positive.
If the original number is less than 1, the count is
negative.
3) Multiply the new number in Step 1 by 10
raised to an exponent equal to the count found
in Step 2.
Example
Write each of the following in scientific notation.
1)
4700
You must move the decimal 3 places to the left, so that
the new number has a value between 1 and 10.
Since we moved the decimal 3 places, and the original
number was > 10, our count is positive 3.
4700 = 4.7  103
2)
0.00047
Have to move the decimal 4 places to the right, so that
the new number has a value between 1 and 10.
Since we moved the decimal 4 places, and the original
number was < 1, our count is negative 4.
0.00047 = 4.7  10-4
To Write a Scientific Notation Number in
Standard Form
•
Move the decimal point the same number of
spaces as the exponent on 10.
•
•
If the exponent is positive, move the decimal point
to the right.
If the exponent is negative, move the decimal
point to the left.
Example
Write each of the following in standard notation.
1)
5.2738  103
Since the exponent is a positive 3, we move the decimal 3
places to the right.
5.2738  103 = 5273.8
2)
6.45  10-5
Since the exponent is a negative 5, we move the decimal
5 places to the left.
00006.45  10-5 = 0.0000645
Reminder:
This homework assignment
on section 5.1 is due
at the start of
next class period.
Note: There are 48 problems in
this assignment, but most of them are short.
(This assignment took last semester’s students
just over an hour to complete.)
You may now
OPEN
your LAPTOPS
and begin working on the
homework assignment.