MATH 1030-02/SPRING 2005 – MULTIPLICATION AND POWERS
In mathematics, we often use special symbols and notations to make our formulas more compact and
easy to work with. This is the case with multiplication and powers. Often times, we will be presented with
problems in math were we have to multpily something by itself several times over. For example, we may
want to multiply 5 by itself 7 times. We could write
(1)
5 · 5 · 5 · 5 · 5 · 5 · 5.
The problem with this is that it’s very cluttered. On top of that, it’s hard to tell from formula (1) just
how many fives we’re trying to multiply together without explicitly counting them. To make things clearer,
we use “power notation.” Recall that the natural numbers are all of the counting numbers {1, 2, 3, . . .}.
Power notation just means that if a is any number and n is a natural number then
n times
}|
{
z
a = a · a · a···a.
(2)
n
By the way, we call the number n the exponent of an .
Using this definition, formula (1) becomes
57
(3)
Now, it should be clear that for m and n natural numbers we have
(4)
m times
m+n times
n times
}|
{ z
}|
{ z
}|
{
z
an · am = a · a · a · · · a · a · a · a · · · a = a · a · a · · · a = am+n .
This is the famous identity that tells us when multiplying powers we add the exponents.
Also, we have that
(5)
(a )
n m
m times
}|
{
z
n
= a · an · an · · · an = amn .
This is the (other) famous identity that tells us when taking the power of a power we multiply exponents.
(We will see that formulas (4) and (5) hold whenever n is any whole number. i.e., n could be zero or negative
and the formulas still work.)
In the business of multiplying, 1 is a very special number. What makes 1 so special? Well, if x is any
number at all x · 1 = 1 · x = x, i.e. multiplication by 1 doesn’t change anything!! This fact is used over and
over (and over and . . . ) in math. In algebra we not only need to know how to multiply numbers, but we
need to perform the “opposite” operation as well. That is, we need to know how to divide numbers.
1
2
MATH 1030-02/SPRING 2005 – MULTIPLICATION AND POWERS
Let’s look closely at an example. Suppose we begin with the equation
(6)
3x = 17.
In order to solve for x we have to apply the opposite operation to multiplying by 3: we must divide by 3 (or
multiply by
1
3 .)
We call
1
3
the multiplicative inverse of 3, and we define a power notation that represents
this new idea:
a−1 =
(7)
1
a
This notation is motivated by formula (4). Notice that
(8)
a1 · a−1 = a · a−1 = a ·
a
1
= = 1 = a0 = a1+(−1) .
a
a
An important thing to notice is that in the above formula we have a0 . From now until the end of time,
one should remember that if a is any number besides zero then a0 = 1. This rule will always be true.
By combining formulas (5) and (7) we get another important property:
a
(9)
=
a1
=
a(−1)(−1)
=
(a−1 )−1
=
1
by (5)
by (7).
a−1
By playing around with this formulas in a similar way we can also see that
(10)
(11)
an
= an−m
am
a n
b
and
=
an
.
bn
In short: all of the boxed formulas could easily be memorized (and it’s not a bad idea to do that), but
the purpose of these notes and the accompanying lecture is to show that there is a logical reason for all of
these formulas and they are not meant to make life more difficult. Actually, besides being very beautiful
properties, they allow us to solve real world problems with much more ease. Also, there are many different
correct ways of approaching exponents and power notation. If you’re having trouble finding a way that works
for you please come by my office and we can try to figure this stuff out in a way that works for you!