find the nth roots

Question

Find all complex square roots of −4-4sqrt3 i . In other words, find all complex solutions of x^2=-4-4sqrt3 i

 

Answer 1

First, put -4 - 4√3 i in polar form r(cosθ + isinθ)

r = √ [(-4)² + (-4√3)²] = √64 = 8

The point (-4, -4√3) lies in the third quadrant. The angle with vertex (0,0) which has initial side along the negative x-axis and terminal side containing the point (-4, -4√3) has measure π/3, So, θ = π + π/3 = 4π/3.

Polar form: 8(cos4π/3 + isin4π/3)

By Demoivre's Theorem, to find one of the square roots, raise 8 to the 1/2 power and divide 4π/3 by 2.

We get √8[cos2π/3 + isin2π/3] = 2√2[ -1/2 + (√3/2)i] = -√2 + √6i

The other square root is √8[cos(2π/3 + π) + isin(2π/3 + π)] = 2√2[1/2 - (√3/2)i] = √2 - √6i

Answer 2

x^2=-4-4*(3i)^(.5)

=-4-4*1.732(i^.5)

=-4-6.928(i^.5)

i=-1^.5