Question

Find five roots of this rectangular or polar expression.
$3+3i\sqrt{3} , n=5$

Answer 1

Answer:

  see below

Step-by-step 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

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

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)