12.1  Exponents
Section 3.3 discussed integer exponents.
This section continues from there and explains noninteger exponents.
We saw that if n is a natural number (i.e. 1, 2, 3, …)
then the exponential b^{ n}
is defined to mean multiplying b by itself n times, like this:
The number b is called the base, n is called the exponent,
and we say that we are raising b to the n^{ th} power.
We also saw that any base raised to a negative integer power is the reciprocal of the
same base raised to the corresponding positive power:
and that any base raised to the 0^{th} power equals 1:
b^{ 0} = 1.
In that section we also saw that exponentials have these three properties:
Multiplication property


Division property


Exponentiation property


Let's assume that these properties can be generalized to exponents that are
not necessarily integers. We will find that we can also give
meaning to rational and real exponents and even complex exponents.
The meaning of b^{ 1/n} :
Let m = 1/n in the exponentiation property. This gives:
On the other hand taking the n^{th} power of the
n^{th} root of b gives the same result:
Putting this together we find that:
In other words, b^{ 1/n} is the
n^{th} root of b. The most important case is:
In other words, b^{ 1/2} is the square root of b.
Note: If we are working over the real numbers and the
base b is negative then n must be an odd integer;
otherwise we can't take the n^{th} root of b.
If we are working over the complex numbers then there is no problem
and no restriction on n.
Click here for more information.
For example the following exponential gives a real value:
but this exponential gives a complex value:
The meaning of b^{ m/n} :
Using the exponentiation property we can write
b^{ m/n} two ways:
But in the previous section we saw that
b^{ 1/n} is the n^{th} root of b.
Thus this equation says that b^{ m/n} may be thought of as the
n^{th} root of the m^{th} power of b
or as the m^{th} power of the n^{th} root of
b.
For example:
A note on negative bases: If we are working over the real numbers and the base b
is negative then n must be an odd integer;
otherwise we can't take the n^{th} root of b.
If we are working over the complex numbers then there is no problem and
no restriction on n.
Click here for more information.
The meaning of b^{ r} where r is any
real number:
We have made sense of exponents that are positive or negative or fractions
but what about exponents that are real numbers?
We will explain this case using an example.
We will show that 10^{ 1.2} equals 16 (approximately).
To do this we make a graph of the function
y = 10^{ x}.
Here is the table of values and the graph with a smooth curve interpolated
through the points.
We see that the interpolated curve goes through the point
( x = 1.2, y = 16 ).
It is in this sense that we can say that 10^{ 1.2} = 16.
It is easy to see that this use of interpolation doesn't depend on the base being 10;
any other base would produce a similar result.
Note: If we are working over the real numbers then the base b
must be a positive number. If we are working over the complex numbers
then the base can have any value, with one exception
(namely the base can't be zero if the real part of the exponent is negative or zero,
because this causes division by zero; see below.)
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The meaning of 0^{ x} :
This is an interesting situation because depending on the exponent x,
there is zero either in the numerator or the denominator,
so the expression can be either 0 or undefined.
And 0^{ 0} cannot be computed; it is defined to equal 1.
In summary, if the exponent x is a real number then there are the
following cases:
If the exponent x is a complex number then there are the following cases: