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Chapter 9: Q. 9 (page 591)

Every nonzero complex number has exactly _______ distinct cube roots.

Short Answer

Expert verified

Every complex number has three cube roots of unity.

Step by step solution

01

Step 1. Given information

An incomplete statement is given as

Every nonzero complex number has exactly _______ distinct cube roots.

02

Step 2. Explanation

Let non zero complex number be $w = r (\cos θ_{0} + i \sin θ_{0})$

And the cube root for this complex number is

role="math" localid="1646734334742" $z_{k} = \sqrt[n]{r} [\cos (\frac{θ_{0}}{n} + \frac{2 k π}{n}) + i \sin (\frac{θ_{0}}{n} + \frac{2 k π}{n})]$

Where k can be any positive integer.

For $n = 3$ we have $k = 0, 1, 2$

As ktakes three values then we have exactly three cube roots.

Therefore every complex number has three cube roots of unity.

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