Exponents

Definition

In xⁿ, x is the base, and n is the exponent (or power)

We defined positive integer powers by

xⁿ = x ^· x ^· x ^· . . . ^· x (n factors of x)

Properties

The above definition can be extended by requiring other powers (i.e. other than positive integers) to behave like the positive integer powers. For example, we know that

xⁿ x^m = x^{n + m}

for positive integer powers, because we can write out the multiplication.

Example: 

            x² x⁵ = (x ^· x)(x ^· x ^· x ^· x ^· x) = x ^· x ^· x ^· x ^· x ^· x ^· x = x⁷

We now require that this rule hold even if n and m are not positive integers, although this means that we can no longer write out the multiplication (How do you multiply something by itself a negative number of times? Or a fractional number of times?).

We can find several new properties of exponents by similarly considering the rule for dividing powers:

(We will assume without always mentioning it that  x ¹ 0). This rule is quite reasonable when m and n are positive integers and m > n. For example:

where indeed 5 – 2 = 3.

However, in other cases it leads to situation where we have to define new properties for exponents. First, suppose that m < n. We can simplify it by canceling like factors as before:

But following our rule would give

In order for these two results to be consistent, it must be true that

or, in general,

·        Notice that a minus sign in the exponent does not make the result negative—instead, it makes it the reciprocal of the result with the positive exponent.

Now suppose that n = m. The fraction becomes

,

which is obviously equal to 1. But our rule gives

Again, in order to remain consistent we have to say that these two results are equal, and so we define

            x⁰ = 1

for all values of x (except x = 0, because 0⁰ is undefined)

 

Summary of Exponent Rules

The following properties hold for all real numbers x, y, n, and m, with these exceptions:

1.      0⁰ is undefined

2.      Dividing by zero is undefined

3.      Raising negative numbers to fractional powers can be undefined

x1 = x	(xn)m = xnm
x0 = 1
xn xm = xn + m