In x^{n}, x is the base, and n is the exponent (or power)
We defined positive integer powers by
x^{n} = x ^{·} x ^{·} x ^{·} . . . ^{·} x (n factors of x)
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
x^{n} x^{m} = x^{n }^{+ m}
for positive integer powers, because we can write out the multiplication.
Example:
x^{2 }x^{5} = (x ^{·} x)(x ^{·} x ^{·} x ^{·} x ^{·} x) = x ^{·} x ^{·} x ^{·} x ^{·} x ^{·} x ^{·} x = x^{7}
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
x^{0} = 1
for all values of x (except x = 0, because 0^{0} is undefined)
The following properties hold for all real numbers x, y, n, and m, with these exceptions:
1. 0^{0} is undefined
2. Dividing by zero is undefined
3. Raising negative numbers to fractional powers can be undefined
x^{1} = x |
(x^{n})^{m} = x^{nm} |
x^{0} = 1 |
_{} |
x^{n} x^{m} = x^{n }^{+ m} |
_{} |
_{} |
_{} |