R.2
Integer Section Exponents, Scientific Notation,
Order of Operations
W hen you see a term like x7 , x is called the base and
7 is called the exponent or power.
1. W hen the exponent is a positive integer (like 7 is), the
term has a simple meaning:
x7
x x x x x x x
7 factors
2. When the exponent is zero and the base is not zero,
then x0 1. Similarly, 30 1, 0 1, 4.20 0 , etc.
Notice we said that the base is not zero,
this is because 00 is undefined.
3. When the exponent is a negative integer, a term like
x 7 1 . Similarly, 1 3 x3 .
7
x
x
Simply stated, to get rid of a negative on an exponent,
you move the term to the other side of the fraction bar
(numerator to denominator or denominator to numerator)
and remove the minus sign.
For example:
x3 y4
y4 x3
1
Math 1513 Sec R.2
Properties of Exponents
Examples
m
n
a a a
a. x4 x6 x46 x10
mn
3 5
2
235
b. 4 4 4 4
9
a. x5 x95 x4
m
a amn (a 0)
n
a
x
6
b. z 5 z 65 z 11 1
z
z11
32
2
n
(am) a
m
a. (53) 5
mn
b. ( y3)4 y
a
b
m
mm
12
1
y12
4
4
4
b. (3x2) 3 ( x2) 81x8
a.
m
am
b
6
5
a. (2 y)3 23 y3 8 y3
(ab) a b
4
4
(b 0)
b.
2
2
3
2
2
4
22 9
3x2
4 y3
3
4
34( x2)4
44( y3)4
81x8
256 y12
Math 1513 Sec R.2
Scientific Notation
As the name suggests, this method for representing
numbers is used in science e.g. physics and chemistry.
It is a shorthand for representing very large numbers or
very small numbers with lots of decimals.
Heres how and why it works:
If you multiply a number by a power of 10, the effect is to
move the decimal to the right the same number of places
as the exponent. For example:
1
2
3
14310 1430 , 14310 14300 , 14310 143000
If you multiply a number by a negative power of 10, the
effect is to move the decimal to the left the same number
of places as the exponent. For example:
1
2
3
14310 14.3, 1.43 14310 , 14310 .143
In scientific notation, we usually want to end up with one
number to the left of the decimal.
___
___
126000 1.2610
0.0000945 9.4510
4
4
4.0510
6.1510
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Math 1513 Sec R.2
More examples:
4
2
1. 1.210 2.410
5
2. 3.84 10
2
3.2 10
4
Math 1513 Sec R.2
Order of Operations
W hen a mathematical term has several operations
(exponentiation, multiplication, division, addition, subtraction)
that must be performed, there are rules that determine
the order in which the operations must be done.
One way to remember this order, is to remember the
initials P.E.M.D.A.S.
This means do things in parentheses first, then the
exponents, then multiplication, then division, then
addition and finally subtraction.
You may have heard of some of these ways to help
remember PEMDAS or you may want to make up your
own.
People Eat More Donuts After School.
Please Excuse M y Dear Aunt Sally.
Purple Elk May Destroy A School.
Pink Elephants Make Delicious Apple Sauce.
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Math 1513 Sec R.2
Since we will be using the calculator to do many of our
calculations, it is important to know the proper calculator
language. Your calculator knows PEMDAS, but many
times you have to help it by inputting additional notation
so that it can do the arithmetic properly.
Example:
Compute:
23 4
7 3
You might be tempted to input this into your
calculator as:
That would be wrong. Entered that way, you are asking
your calculator to multiply 2x3 then divide 4 by 7. Take
those two results add them together and subtract 3.
But in the original problem, the numerator 23 4 is to be
divided by the denominator 7 4 .
This would be: 23 4 6 4 2 0.666...
74
3
3
Probably the best and fastest way to work this problem in
one step on the calculator would be:
Make sure you understand why we needed to put
parentheses around the numerator and denominator.
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Math 1513 Sec R.2