Exponents
Definition
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)
Properties
The above definition can be extended by requiring other powers (i.e. otherthan 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 notpositive integers, although this means that we can no longer write out themultiplication (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 makethe 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)
Summary of Exponent Rules
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} |
_{} |
_{} |
_{} |