Question 211188
determine the number of roots of unity? The solution for the following problems must be made using the radian not polar rectangular approach: 
(1-i)^1/4
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On a coordinate system, plot the point (1,-1)
Notice that it is in the 4th quadrant.
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r = sqrt(1 + 1) = sqrt(2)
theta = arctan(-1/1) = 315 degrees in the 4th quadrant.
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(1-i)^(1/4) = r^(1/4)(cis[(315+(n(360)]/4) where n = 0,1,2,3
= 2^(1/8)cis(78.75 degrees) when n=0
= 2^(1/8)cis(78.75+90) when n = 1
= 2^(1/8)cis(78.75+180) when n = 2
= 2^(1/8)cis(78.75+270) when n = 3
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(1+i*sqrt(3)^7/2
Plot the point; it is in the 1st quadrant.
r = sqrt(1+3) = 2
theta = arctan(sqrt(3)) = 60 degrees
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(1+i*sqrt(3))^(7/2) = 2^(7/2)cis[(60 + n*360)]/(7/2) when n = 0,1. etc.
= 2^(7/2)cis(17.14 degrees) when n = 0
= 2^(7/2)cis(17.14+102.86) when n = 1
etc.
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(-i)^1/2 
Plot the point (0,-1)
r = sqrt(0+1) = 1
theta = arctan(-1/0) which is undefined but you can see that theta = 270
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(-i)^(1/2)= 1cis(270+360n)/2
= cis(135) when n=0
= cis(135+180) when n = 1
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Cheers,
Stan H.