Chapter 6.5, Problem 3E

To determine

Tofind:the complex number $\text{u=a+bi}$ is an_______ of the complex number z when ${\text{z=u}}^{\text{n}}$ .

Answer to Problem 3E

The nth root of complex number

Explanation of Solution

Given:

  $\text{u=a+bi}$

  ${\text{z=u}}^{\text{n}}$

Concept used:

There is ${\text{n}}^{\text{th}}$ distinct roots corresponding to

  $\text{k=0,1,2,}\mathrm{...........}\text{,n-1}$

Therefore the ${\text{n}}^{\text{th}}$ root of the unity is

  ${\text{w}}_{\text{n}}\text{=exp}\left(\frac{\text{2πk}}{\text{n}}\right)$ .

Calculation:

The complex number $\text{u=a+bi}$ is an nth root of complex number z if

  ${\text{z=u}}^{\text{n}}$

  $\Rightarrow {\left(\text{a+bi}\right)}^{\text{n}}$

Let’s consider the complex number $z=-1$ .

Or let’s find the second roots of this complex number.

It can be written as:

  $z=-1$ .

After taking the square root both side of the above equation:

  $\sqrt{z}=\pm \sqrt{-1}$

  $\Rightarrow \text{±i}$

Hence, the complex number $\text{0+i and 0-i}$ are the two second root of the complex number $\text{-1}\text{.}$