Question

Find (√3+i)^1/3

Answer 1

not sure if there's anything to find there, or even to simplify

Answer 2

\[Let x=\left( \sqrt{3}+\iota\right) =2\left( \frac{ \sqrt{3} }{2 }+\frac{ 1 }{2 }\iota \right) =2\left( \cos \frac{ \pi }{ 6 }+\iota \sin \frac{ \pi }{ 6 } \right)\]
\[x=2 e ^{2n \pi+\frac{ \pi }{6 }}\]
\[x ^{\frac{ 1 }{ 3 }}=2^{\frac{ 1 }{3 }}e ^{\frac{ 2n \pi+\frac{ \pi }{6 } }{3 }}\]
substitute n=0,1,2 and get the reqd result.

Answer 3

correction \[x=2^{\frac{ 1 }{3 }} e ^{\iota \left( 2n \pi+\frac{ \pi }{ 6} \right)}\]