Question

Find all values of `(1 + i)^(1/3` and show that their continued product is (1+ 𝒊 ).

Answer 1

Let Z =`(1 + i)^(1/3` 

`Z= [sqrt2(cos  pi/4+isin  pi/4)]^(1/3)`

`Z = (sqrt2)^(1/3).[cos(2kpi+pi/4)+isin(2kpi+pi/4)]^(1/3`

`Z=2^(1/6)[cos(2kpi+pi/4)+isin(2kpi+pi/4)]^(1/3)`

Putting k = 0, 1, 2.

`Z_0=2^(1/6).e^((ipi)/12)`

`Z_1=2^(1/6).e^((9ipi)/12)`

`Z_2=2^(1/6).e^((17pi)/12)`

`therefore Z_0 Z_1 Z_2=2^(3/6).e^((27pi)/12)`

`=2^(1/6).e^((9ipi)/4)`

`=sqrt2(cos  (9pi)/4+isin  (9pi)/4)`

`=sqrt2(1/sqrt2+i1/sqrt2)`

= (1 + i).