Question

Please find the square roots of the following complex number.


`z = 81(\cos((4\pi)/(9))+i\sin((4\pi)/(9)))`

Answer 1

Hey there,

First, find the square root of the given equation.

`√(z)=\sqrt{81(\text{cos}((4\pi)/(9))+i\text{sin}((4\pi)/(9))}\\=9\text{cos}(1)/(2)(2k\pi+((4\pi)/(9)))+i\text{sin}(1)/(2) (2k\pi+((4\pi)/(9)))`

Keep in mind that this is solved for k = 0 and 1.

Second, simplify.

`=9\text{cos}((2\pi)/(9))+i\text{sin}((2\pi)/(9))~\text{or}~9\text{cos}((11\pi)/(9))+i\text{sin}((11\pi)/(9))`

Best of Luck!