Let
(1) Natural Powers
If
(2) Zero Power
If
00 is not defined.
(3) Rational Powers
If
We need to consider two cases.
-
- Case 1: n is a positive odd integer. There is exactly one nth root for each real b.
-
Case 2:
is a positive even integer:
Case 2a: If b < 0 , there is no nth root.
[If we multiply a positive or negative number by itself an even number of times the result will be positive (for example, (-2)4=32>0). So there cannot be a real number u such that un=b as un ≥ 0 for all real u and positive even integer n.]
Case 2b: If b > 0, there are two nth roots (one positive and one negative).
However, the symbol b1/n or
[There are two nth roots because (-u)n=un when n is even; that is, for each b>0 there will be two nth roots, u and -u.]
Remark:
-
if if and is odd is imaginary ( = not real) if and is even.
Properties of the nth roots
Let a and b be two real numbers and m and n be two integers. We can show that the nth roots have the following properties.
(a)
(b)
(c)
(d)
- Note that when
is even and both and , then and exist (they are real numbers) because and , but and do not exist (in fact, they are imaginary).
If
We can verify that also
If n is even, we require that b > 0.
(4) Irrational Powers
If
Because we can write
- Note that when
is irrational, is imaginary for b < 0 .
(5) Negative Powers
For any real number
For example,
General Properties
If
(a)
(b)
(c)
(d)
(e)