How do you simplify (-1)^(-1/8)?

2 Answers
Dec 28, 2015

Deleted answer.

Dec 28, 2015

(-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(pi/8) and sin(pi/8), which we use later...

cos^2(theta/2) = 1/2 (1 + cos(theta))

sin^2(theta/2) = 1/2 (1 - cos(theta))

Let theta = pi/4.

Then cos(theta) = sqrt(2)/2 and we find:

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 pi/8 is in Q1 we want the positive square roots:

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: e^(i pi) = -1

So (-1)^(1/8) = (e^(i pi))^(1/8) = e^(i pi/8) = cos(pi/8) + i sin(pi/8)

There are 7 other 8th roots of -1, but this is the principal one.

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