Question 681027
<pre>
{{{z^4}}}{{{""=""}}}{{{-2 + 2sqrt(3)i}}}

{{{matrix(2,1,"",z)}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",(2 + 2sqrt(3)i)^(1/4))}}}

{{{x = -2}}}, {{{y =2sqrt(3)}}}, {{{r=sqrt(x^2+y^2)=sqrt((-2)^2+(2sqrt(3))^2)}}} = {{{sqrt(4+4*3)}}} = {{{sqrt(4+12)}}} = {{{sqrt(16)}}} = {{{4}}}
 
{{{tan(theta)}}}{{{""=""}}}{{{y/x}}}{{{""=""}}}{{{(2sqrt(3))/(-2)}}}{{{""=""}}}{{{-sqrt(3)}}}.

The reference angle for {{{theta}}} is 60° and since x is - and y is +,
{{{theta}}} is in the 2nd quadrant, so we subtract 60° from 180° and get
120°, which has the same trig functions as 120° + 360°·k, so we take:

{{{theta}}}{{{""=""}}}{{{"120°"+"360°"k}}}.

Now we have

{{{matrix(2,1,"",z)}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",(2 + 2sqrt(3)i)^(1/4))}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",(r(cos(theta)+i*sin(theta)))^(1/4))}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",(4(cos("120°"+"360"k)+i*sin("120°"+"360"k)))^(1/4))}}}

Now we use the formula: {{{(r(cos(theta)+i*sin(theta)))^n}}}{{{""=""}}}{{{r^n*(cos(n*theta)+i*sin(n*theta))}}}

{{{matrix(2,1,"",z)}}}{{{matrix(2,1,"",""="")}}}{{{matrix(2,1,"",4^(1/4)(cos(expr(1/4)("120°"+"360"k))+i*sin(expr(1/4)("120°"+"360"k))))}}}{{{matrix(2,1,"",""="")}}}

{{{matrix(2,1,"",(2^2)^(1/4)(cos("30°"+"90"k)+i*sin("30°"+"90"k)))}}}

{{{matrix(2,1,"",(2^(1/2))(cos("30°"+"90"k)+i*sin("30°"+"90"k)))}}}{{{matrix(2,1,"",""="")}}}
{{{sqrt(2)(cos("30°"+"90"k)+i*sin("30°"+"90"k))}}}

Now we let k = 0,1,2,and 3 to get the 4 4th roots, the 4 possible
answers for z:

For k = 0,

{{{sqrt(2)(cos("30°"+"90"*0)+i*sin("30°"+"90"*0))}}}{{{""=""}}}{{{sqrt(2)(cos("30°")+i*sin("30°"))}}}

For k = 1,

{{{sqrt(2)(cos("30°"+"90"*1)+i*sin("30°"+"90"*1))}}}{{{""=""}}}{{{sqrt(2)(cos("120°")+i*sin("120°"))}}}

For k = 2,

{{{sqrt(2)(cos("30°"+"90"*2)+i*sin("30°"+"90"*2))}}}{{{""=""}}}{{{sqrt(2)(cos("210°")+i*sin("210°"))}}}

For k = 3,

{{{sqrt(2)(cos("30°"+"90"*3)+i*sin("30°"+"90"*3))}}}{{{""=""}}}{{{sqrt(2)(cos("300°")+i*sin("300°"))}}}

Those are the four values for z.

Edwin</pre>