Find 4 values of following in exponential forms ?

Find 4 values of following in exponential forms ?

(sqrt3 -i)^(1/4)

1 Answer
Jan 17, 2018

See below

Explanation:

let
sqrt3-i=x^4

x^4 = 2(sqrt3/2-i/2)

x^4 = 2[cos(-pi/6) + isin(-pi/6)]

x^4 = 2[cos(2mpi - pi/6) + isin(2mpi - pi/6)]

(This small addition will not the change the value of the equation. Check out yourself.)

x = 2^(1/4)[cis(2mpi-pi/6)]^(1/4)

Short form of writing costheta + isintheta = cistheta

x=2^(1/4)[cis((2mpi)/4 - (pi/6)*1/4)] (applying Demoveries theorm)

x = 2^(1/4)[cis((mpi)/2-pi/24)]

Since the equation was of degree 4, there will be only four roots.

1st root
putting m =0

x = 2^(1/4)[cis(-pi/24)] = 2^(1/4)*e^(-ipi/(24))

2nd root
putting m =1

x = 2^(1/4)[cis((11pi)/24)] = 2^(1/4)*e^(i(11pi)/(24))

3rd root
putting m =2

x = 2^(1/4)[cis((23pi)/24)] = 2^(1/4)*e^(i(23pi)/(24)

4th root
putting m =3

x = 2^(1/4)[cis((35pi)/24)] = 2^(1/4)*e^(i(35pi)/(24))