Find 4 values of following in exponential forms ?

Question

Find 4 values of following in exponential forms ?

#(sqrt3 -i)^(1/4)#

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Answer 1 · 2018-01-17T12:14:12

See below

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))#