x81 solve for x x81 solve for x x81 solve for xSolutionx8 1

Solution

x^8 -1 =0 solve for x

Find 8th root of unity :

Use the polar form of the number with De Moivre\'s Theorem.

x^8 = 1 = cos 0 + i sin 0
= cos(2 pi k) + i sin(2 pi k) ---- k being any integer.

De Moivre\'s Theorem, the eight roots of 1 are
cos(2 pi k/8) + i sin(2 pi k/8), plug k = 0,1,2,...,8

8th roots of unity are
k = 0: cos 0 + i sin 0 = 1 = x1
k = 1: cos(pi/4) + i sin(pi/4) = sqrt(2)/2 + i sqrt(2)/2 = x2
k = 2: cos(pi/2) + i sin(pi/2) = i = x3
k = 3: cos(3pi/4) + i sin(3pi/4) = -sqrt(2)/2 + i sqrt(2)/2 = x4
k = 4: cos(pi) + i sin(pi) = -1 = x5
k = 5: cos(5pi/4) + i sin(5pi/4) = sqrt(2)/2 - i sqrt(2)/2 = x6
k = 6: cos(3pi/2) + i sin(3pi/2) = -i = x7
k = 7: cos(7pi/4) + i sin(7pi/4) = -sqrt(2)/2 - i sqrt(2)/2 = x8

x1, x2.... x8 are roots of unity