Question

Find five roots of this rectangular or polar expression.

`3+3i√(3) , n=5`

Answer 1

see below

Explanation:

The n-th root of a complex number is the n-th root of its magnitude at 1/n times the angle (plus 0 to n-1 multiples of 2π/n radians).

Here, the magnitude is 6 and the angle is π/3

`\sqrt[5]{3+3i√(3)}=(6\angle{((\pi)/(3)}+2n\pi))^{(1)/(5)}=\sqrt[5]{6}\angle\pi\{(1)/(15),(7)/(15),(13)/(15),(19)/(15),(5)/(3)\}`

In terms of reference angles, these are ...

• (6^(1/5))(cos(π/15) +i·sin(π/15))
• (6^(1/5))(sin(π/30) +i·cos(π/30))
• (6^(1/5))(cos(2π/15) +i·sin(2π/15))
• (6^(1/5))(sin(7π/30) +i·cos(7π/30))
• (6^(1/5))(1/2 -i·(√3)/2)