Chapter 14 - Introduction to Exponents
### Chapter 14 - Introduction to Exponents

This chapter on exponents marks the
beginning of the second half of our study of Calculus. In the first half
we
introduced the concepts of numbers, functions and graphs, then went on to
analyze them in more depth using the derivative. Having understood what
the
derivative was, the remainder of the first half was devoted to
integration, or
the reverse of differentiation. It required some abstract thinking since
what
we were doing was analyzing that which we could net directly see.

The first half we basically focused
entirely on simple rules of arithmetic. Differentiation was presented as a
subtracting and dividing process while Integration was shown to be a
multiplicative and additive operation. In other words our study of
Calculus was
limited to multiplication and addition. From the first chapter you might
recall
how multiplication was defined as
repetitive addition where x multiplied by y was just x plus itself
y
times. With this idea we can say that all multiplication reduces to
addition
and this means our study of Calculus was based entirely on addition.

Now we will study Calculus based on
multiplication through the study of exponential and Logarithmic function.
Before continuing we need to establish a few properties of exponents. The
rules
that govern exponents are very similar to addition. The definition of an
exponent or a power is:

Where a is
called the base, and x is the exponent or power. Therefore a raised to the
x
power is defined to be:

For example:

Compare this
with the definition of multiplication where:

Since
multiplication can be called repetitive addition then exponent or raising
something to a power can be though of as repetitive multiplication. This
is
really all that we mean by exponents. They can be reduced to
multiplication and
multiplication can be further reduced to exponents.

Now let us take a look at some
properties of exponents. The first being the rule for multiplying two
exponents
of the same base.

Notice how
multiplication of exponents has been reduced to addition of exponents. For
example:

It is really
this simple. The second important property of exponents states that a to
the x
power raised to the y power is just a to the product of x and y power
or:

All this is saying is that an exponent raised to another
exponent can be reduced to multiplying the two exponents out or:

Take note of how
this rule reduces exponents of exponents to just multiplication of the two
exponents whereas rule one reduces multiplication of exponents to addition
of
exponents, ( assuming a common base a).

These two important properties of
exponents are the fundamental ways of defining what exponents are and how
they
relate to repetitive multiplication, where multiplication is just
repetitive
addition. It now remains for us to define what we mean by raising an
exponent
to a fractional power. This is actually much simpler than it sounds.

Since we defined whole number
exponents to denote repetitive multiplication, we would want fractional
powers
to represent repetitive **division.**
Fractional exponents therefore are called roots and tell us into how many
equal
multiples a number has been divided into ,such that the product of these
roots
gives us the original number a back.

This definition
of fractional exponents remains consistent with the first and second rule
of
adding and multiplying exponents.

Last but not least we need to define
what we mean by raising exponents to negative numbers. This is done by
remembering that positive exponents
refer to multiplication, therefore negative exponents would refer
to
division. In order for our definition to remain in accordance with the two
rules of exponents we need to define it this way:

This is rather a
long way of deriving this as you can clearly see from the following
example how
negative exponents are defined.

Increasing the power means
multiplying it more times by itself, while decreasing the power means
dividing
it more times by itself. Therefore multiplication and division reduce to
addition and subtraction in exponents.

Since a raised to the first power is
a multiplied by itself once, or just a, by definition then what is a
raised to
the zero power. Once again this definition must remain consistent with the
rules of exponents. Instead of raising a to zero power, we can raise it to
1/∞ or one over infinity which is close to zero. Therefore a raised
to
the negative 1 over infinity is:

From the definition
of fractional exponents, we are
asking
ourselves what number multiplied by itself an infinite amount of times
gives
a? If this number were slightly less than 1, then as you multiply infinite
times, you get a smaller and smaller number or zero. If this number were
slightly greater than 1 then you would get a larger and larger number each
time
you multiply it by itself, or eventually infinity. Therefore only the
number
one satisfies both limits as you approach it from either 1/∞ or
-1/∞. Hence:

You might think
that how can:
. This is because zero and 1/∞ are not exactly the
same.

This definition of zero exponent
power tells us an important property of exponential.s. In multiplication
you
are adding, where adding nothing is zero. In exponents you are
multiplying,
where multiplying nothing is one, not zero. Therefore all exponents are
expressed as repetitive multiplication of numbers greater than or equal to
one.
We can say that the base of exponents is therefore 1.