Back to Mth Roots
As mentioned in §
3.4, there are

different numbers

which satisfy

when

is a positive integer.
That is, the

th root of

, which is
written as

, is not unique--there are

of them. How do
we find them all? The answer is to consider
complex numbers in
polar form.
By
Euler's Identity, which we just proved, any number,
real or complex, can be written in polar form as

where

and

are
real numbers.
Since, by Euler's identity,

for every integer

, we also have
Taking the

th root gives
There are

different results obtainable using different values of

,
e.g.,

. When

, we get the same thing as
when

. When

, we get the same thing as when

, and so
on, so there are only

distinct cases. Thus, we may define the

th

th-root of

as
These are the

th-roots of the
complex number

.
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