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To show the complex nth roots of nonzero
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Answer to Problem 65AYU
the complex nth roots of a nonzero complex number w are equally spaced on the circle.
Explanation of Solution
Let be any non-zero complex number and n be any positive integer greater than or equal to 2.
The distinct complex roots of
are
, where
is a complex number,
and
Here the argument of the complex number is and varies as the value of k changes. Thus, the arguments of the complex roots obtained are equally spaced. Also, it is known that, the complex roots obtained from this formula when plotted on a graph form a circle with centre at origin and radius equal to the magnitude
Hence it is proved that, the complex nth roots of a nonzero complex number w are equally spaced on the circle.
Chapter 9 Solutions
Precalculus
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