Question

Every nonzero complex number has exactly _______ distinct cube roots.

Short answer

Every complex number has three cube roots of unity.

Step-by-step solution

Step 1. Given information

An incomplete statement is given as

Every nonzero complex number has exactly _______ distinct cube roots.

Step 2. Explanation

Let non zero complex number be $w=r\left(\cos {\theta }_0+i\sin {\theta }_0\right)$

And the cube root for this complex number is

role="math" localid="1646734334742" $z_k=\sqrt[n]{r}\left[\cos \left(\frac{{\theta }_0}{n}+\frac{2k\pi }{n}\right)+i\sin \left(\frac{{\theta }_0}{n}+\frac{2k\pi }{n}\right)\right]$

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.