Question 565116
<pre>
You have to know 2 things:

1. How to change from a+bi form to trig form r(cos<font face = "symbol">q</font> + i sin<font face = "symbol">q</font>)

use tan<font face = "symbol">q</font> = {{{b/a}}} and r = {{{sqrt(a^2+b^2)}}}

2.  How to raise a complex number in trig form using DeMoivre's
   formula:

[r(cos<font face = "symbol">q</font> + i sin<font face = "symbol">q</font>)]<sup>n</sup> = r<sup>n</sup>[cos(n<font face = "symbol">q</font>) + i·sin(n<font face = "symbol">q</font>)]


(2+2squareroot3i)^6


1. To change from 2+2{{{sqrt(3)}}}i form to trig form r(cos<font face = "symbol">q</font> + i·sin<font face = "symbol">q</font>)

use tan<font face = "symbol">q</font> = {{{(2sqrt(3))/2}}} = {{{sqrt(3)}}}
therefore <font face = "symbol">q</font> = 60°

and r = {{{sqrt(2^2+(2sqrt(3))^2)}}} = {{{sqrt(4+4*3)}}} = {{{sqrt(4+12)}}} = {{{sqrt(16)}}} = 4

So 2 + 2{{{sqrt(3)}}}i = 4(cos60° + i·sin60°) 


2. To raise this complex number in trig form using DeMoivre's
   formula:

[4(cos60° + i·sin60°)]<sup>6</sup> = 4<sup>6</sup>[cos(6·60°) + i·sin(6·60)] =
4096(cos360° + i·sin360°) = 4096(cos0° + i·sin0°) = 
4096(1+i·0) = 4096(1+0) = 4096(1) = 4096.

answer: 4096.

Edwin</pre>