SOLUTION: find the fourth roots of {{{ Z^4=8(-1+sqrt(3)i) }}} giving your answers in the form a+bi where a and b are real numbers. I have gotten to this point,Z^4=16(COS 5/3 PIE+i SIN 5/3

Question 544045: find the fourth roots of giving your answers in the form a+bi where a and b are real numbers.
I have gotten to this point,Z^4=16(COS 5/3 PIE+i SIN 5/3 PIE) i am not sure if am right.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
find the fourth roots of giving your answers in the form a+bi where a and b are real numbers.
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z^4 = -8+8sqrt(3)i
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r = sqrt[8^2 (8sqrt(3))^2] = sqrt[64 + 64*3] = 2^4
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theta = tan^-1(8sqrt(3)/(-8)) = -60 degrees or 120 degrees
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since cos(-60) = cos(60) and since sin(-60) = -sin(60),
z^4 = 2^4(cos(120)+isin(120))
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4th Roots:
z = 2(cos((120+360n)/4) + isin((120+360n)/4))/4
n= 0: z = 2(cis(30) = 2(cos(30)+isin(30)) = 2(sqrt(3)/2 + i/2) = sqrt(3)+i
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n= 1: z = 2(cis(120))= 2(cos(120)+isin(120)) = 2(-sqrt(3)/2+i/2)= -sqrt(3)+i
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n= 2: z = 2(cis(210)) = 2(cos(210)+isin(210)) = -sqrt(3)-i
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n= 3: z = 2(cis(300)) = sqrt(3)-i
=====================================
Cheers,
Stan H.
================
z = 2(cos(30)+ isin(30))
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