Let be a nonzero complex number where
. Then, let
and
and define another complex number
. Now, let
for
.
- Prove that
. We say
is an
th root of
.
- Prove that
has exactly
distinct
th roots given by
and that they are equally spaced on a circle of radius
.
- Find the three cube roots of
.
- Find the four fourth roots of
.
- Find the four fourth roots of
.
- Proof. Using the definitions of
and
we compute,
- Proof. From part (a) we know
is an
th root of
. Then, if
we have
(since
for all
). Hence,
is an
th root of
.
Then, if
and
we have
for
. This implies
for
; hence, there are
distinct values of
. Therefore,
has exactly
distinct
th roots. By the Fundamental Theorem of Algebra we know it cannot have more than
roots; hence, we have shown that they are all of the form
- For the complex number
we have
Therefore, from parts (a) and (b) we have
Hence, the three cube roots of
are
- This time we have
Therefore the four fourth roots of
are
- For
we have
Therefore,
Thus,