How do you simplify (-1)^(-1/8)?
2 Answers
Deleted answer.
(-1)^(-1/8) = sqrt(2+sqrt(2))/2 - sqrt(2-sqrt(2))/2 i
Explanation:
We can use the half angle formulas to find
cos^2(theta/2) = 1/2 (1 + cos(theta))
sin^2(theta/2) = 1/2 (1 - cos(theta))
Let
Then
cos^2(pi/8) = 1/2 (1 + sqrt(2)/2) = (2 + sqrt(2)) / 4
sin^2(pi/8) = 1/2 (1 - sqrt(2)/2) = (2 - sqrt(2)) / 4
So, since
cos(pi/8) = sqrt((2+sqrt(2))/4) = sqrt(2+sqrt(2))/2
sin(pi/8) = sqrt((2-sqrt(2))/4) = sqrt(2-sqrt(2))/2
Note that:
So
There are
So we find:
(-1)^(-1/8)
= 1/(-1)^(1/8)
= 1/root(8)(-1)
= 1/(cos (pi/8) + i sin (pi/8))
=(cos(pi/8) - i sin(pi/8))/((cos(pi/8) - i sin(pi/8))(cos(pi/8) + i sin (pi/8))
=(cos(pi/8) - i sin(pi/8))/(cos^2(pi/8) + sin^2(pi/8))
=cos(pi/8) - i sin(pi/8)
= sqrt(2+sqrt(2))/2 - sqrt(2-sqrt(2))/2 i