Find 4 values of following in exponential forms ?

Question

Find 4 values of following in exponential forms ?

Find 4 values of following in exponential forms ?

${(\sqrt{3} - i)}^{\frac{1}{4}}$ $(sqrt3 -i)^(1/4)$

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Answer 1 · 2018-01-17T12:14:12

See below

Explanation:

let
$\sqrt{3} - i = x^{4}$ $sqrt3-i=x^4$

$x^{4} = 2 (\frac{\sqrt{3}}{2} - \frac{i}{2})$ $x^4 = 2(sqrt3/2-i/2)$

$x^{4} = 2 [cos (- \frac{π}{6}) + i sin (- \frac{π}{6})]$ $x^4 = 2[cos(-pi/6) + isin(-pi/6)]$

$x^{4} = 2 [cos (2 m π - \frac{π}{6}) + i sin (2 m π - \frac{π}{6})]$ $x^4 = 2[cos(2mpi - pi/6) + isin(2mpi - pi/6)]$

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

$x = 2^{\frac{1}{4}} {[c i s (2 m π - \frac{π}{6})]}^{\frac{1}{4}}$ $x = 2^(1/4)[cis(2mpi-pi/6)]^(1/4)$

Short form of writing $cos θ + i sin θ = c i s θ$ $costheta + isintheta = cistheta$

$x = 2^{\frac{1}{4}} [c i s (\frac{2 m π}{4} - (\frac{π}{6}) \cdot \frac{1}{4})]$ $x=2^(1/4)[cis((2mpi)/4 - (pi/6)*1/4)]$ (applying Demoveries theorm)

$x = 2^{\frac{1}{4}} [c i s (\frac{m π}{2} - \frac{π}{24})]$ $x = 2^(1/4)[cis((mpi)/2-pi/24)]$

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

1st root
putting m =0

$x = 2^{\frac{1}{4}} [c i s (- \frac{π}{24})] = 2^{\frac{1}{4}} \cdot e^{- i \frac{π}{24}}$ $x = 2^(1/4)[cis(-pi/24)] = 2^(1/4)*e^(-ipi/(24))$

2nd root
putting m =1

$x = 2^{\frac{1}{4}} [c i s (\frac{11 π}{24})] = 2^{\frac{1}{4}} \cdot e^{i \frac{11 π}{24}}$ $x = 2^(1/4)[cis((11pi)/24)] = 2^(1/4)*e^(i(11pi)/(24))$

3rd root
putting m =2

$x = 2^{\frac{1}{4}} [c i s (\frac{23 π}{24})] = 2^{\frac{1}{4}} \cdot e^{i \frac{23 π}{24}}$ $x = 2^(1/4)[cis((23pi)/24)] = 2^(1/4)*e^(i(23pi)/(24)$

4th root
putting m =3

$x = 2^{\frac{1}{4}} [c i s (\frac{35 π}{24})] = 2^{\frac{1}{4}} \cdot e^{i \frac{35 π}{24}}$ $x = 2^(1/4)[cis((35pi)/24)] = 2^(1/4)*e^(i(35pi)/(24))$

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Find 4 values of following in exponential forms ?