# Calculate in C the ecuation: (1+i√3)z^5=1+i

sciencesolve | Certified Educator

You need to solve for z the given equation, such that:

`(1 + isqrt3)*z^5 = 1 + i`

Isolating `z^5` to one side, yields:

`z^5 = (1 + i)/(1 + isqrt3) => z^5 = ((1 + i)(1 - isqrt 3))/(1^2 - 3i^2)`

`z^5 = (1 - isqrt3 + i + sqrt 3)/(1 + 3)`

`z^5 = (1 + sqrt 3 - i(sqrt3 - 1))/4 => z = ((1 + sqrt 3 - i(sqrt3 - 1))/4)^(1/5)`

You need to use De Moivre's theorem, hence, you need to convert the rectangular form of complex number into polar form, such that:

`z^5 = (|z|(cos alpha + i*sin alpha))`

`|z| = sqrt((1 + sqrt3)^2/16 + (-(sqrt 3 - 1)^2)/16)`

`|z| = sqrt(1 + 2sqrt3 + 3 + 3 - 2sqrt3 + 1)/4`

`|z| = 2sqrt2/4 => |z| = sqrt2/2`

`tan alpha = -(sqrt3 - 1)/(sqrt3 + 1) => tan alpha = -(sqrt3 - 1)^2/2`

`tan alpha = -(3 - 2sqrt3 + 1)/2 => tan alpha = -(2 - sqrt3)`

`alpha = tan^(-1)(sqrt3 - 2)`

`z^5 = (sqrt2/2(costan^(-1)(sqrt3 - 2) + i*sin tan^(-1)(sqrt3 - 2)))`

Using De Moivre's theorem, yields:

`z = (sqrt2/2(costan^(-1)(sqrt3 - 2) + i*sin tan^(-1)(sqrt3 - 2)))^(1/5)`

`z = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 2npi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 2npi)/5)`

You need to give values to n such that:

`n = 0 => z_1 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2))/5 + i*sin (tan^(-1)(sqrt3 - 2))/5)`

`n = 1 => z_2 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 2pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 2pi)/5)`

`n = 2 => z_3 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 4pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 4pi)/5)`

`n = 3 => z_4 = root(5)(sqrt2/2)(cos (tan ^(-1)(sqrt3 - 2) + 6pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 6pi)/5)`

`n = 4 => z_5 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 8pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 8pi)/5)`

Hence, evaluating the complex solutions, using De Moivre's theorem, yields `z_1 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2))/5 + i*sin (tan^(-1)(sqrt3 - 2))/5) ; z_2 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 2pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 2pi)/5) ; z_3 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 4pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 4pi)/5) ; z_4 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 6pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 6pi)/5) ; z_5 = root(5)(sqrt2/2)(cos (tan^(-1)(sqrt3 - 2) + 8pi)/5 + i*sin (tan^(-1)(sqrt3 - 2) + 8pi)/5).`