Chapter 9.3, Problem 63AYU

Show that each complex $n$th root of a nonzero complex number w has the same magnitude.

To determine

To show: Each complex nth root of a nonzero complex number $w$ has the same magnitude.

Answer to Problem 63AYU

Solution:

For a positive integer the complex number has exactly $n$ distinct $n$th roots given by $z_k=\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+ i\sin \frac{\theta +2\pi k}{n}\right)$, where $k=0,1,2\dots$ Here magnitude is $\sqrt[n]{r}$ and angle is $\frac{\theta +2\pi k}{n}$.

From the above formula angle only varies when $k$ values vary and magnitude is same for all the values of $k$.

Hence each complex $n$th root of a nonzero complex number $w$ has the same magnitude.

Explanation of Solution

For a positive integer the complex number has exactly $n$ distinct $n$th roots given by $z_k=\sqrt[n]{r}\left(\cos \frac{\theta +2\pi k}{n}+ i\sin \frac{\theta +2\pi k}{n}\right)$, where $k=0,1,2\dots$ Here magnitude is $\sqrt[n]{r}$ and angle is $\frac{\theta +2\pi k}{n}$.

From the above formula angle only varies when $k$ values vary and magnitude is same for all the values of $k$.

Hence each complex $n$th root of a nonzero complex number $w$ has the same magnitude.