Answer
$\sqrt{3}+i$
Work Step by Step
First, we use the division theorem to divide the absolute values and subtract the arguments:
$\frac{4(\cos150^{\circ}+i\sin150^{\circ})}{2(\cos120^{\circ}+i\sin 120^{\circ})}
\\=2(\cos (150^{\circ}-120^{\circ})+i\sin(150^{\circ}-120^{\circ}))
\\=2(\cos30^{\circ}+i\sin30^{\circ})$
Since we know that $\cos30^{\circ}=\frac{\sqrt{3}}{2}$ and $\sin30^{\circ}=\frac{1}{2}$, we can substitute these values in the expression and simplify:
$2(\cos30^{\circ}+i\sin30^{\circ})
\\=2(\frac{\sqrt{3}}{2}+\frac{1}{2}i)
\\=\sqrt{3}+i$
