Finding all complex fourth roots
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Find all complex fourth-roots in rectangular form of
w= 121 ( cos 2pi/3 + i sin 2pi/3 )
Type answer in the form a + bi round to the nearest tenth.
Zsub 0 = ? + ?i
Zsub 1 = -? + ?i
Zsub 2 = -? - ?i
Zsub 3 = ? - ?i
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Solution Summary
We show how DeMoivre's Theorem can be used to compute all the fourth roots of a number.
Solution Preview
If we write the solution of z^4 = w as:
z = r [cos(theta) + i sin(theta)]
then the equation becomes:
r^4 [cos(4 theta) + i sin(4 theta) ] = 121 [ cos(2pi/3) + i sin(2pi/3) ]
This then implies that:
r = 121^(1/4) = sqrt(11)
and
4 theta = 2 ...
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