Question

How do I find the fifth roots of i?

Answer 1

The answer is in the attachment

Answer 2

Let's do it the hard way. We want to solve the equation x5−1=(x−1)(x4+x3+x2+x+1)=0. Then we are interested in solving x4+x3+x2+x+1=0. Note the symmetry: if r is a root, 1/r is also a root. Why? Thereafter, we have two ways out. First, we divide everything by x2 to get x2+x+1+ 1 x + 1 x2 =(x+ 1 x )+(x+ 1 x )2−1=u2+u−1=0. What did we do? We noted (x+x−1)2−2=x2+x−2 and substituted xu2+u=1+x−1=u. The solutions for u2+u=1 are −φ and φ−1, φ being the golden ratio. We now have two equations: x+ 1 x =−φ⟹x2+φx+1=0⟹x= ± √ φ−3 −φ 2 x+ 1 x =φ−1⟹x2+(1−φ)x+1=0⟹x= ± √ −φ−2 +φ−1 2 . You can now manipulate it to get a a+bi form. Alternatively, you can multiply (x−r)(x−r−1), divide x4+x3+x2+x+1 by it and discover which r makes remainder zero. This would be somewhat long, so I won't explictly do it, but here is the idea. Maybe can help you Or open this web http://math.stackexchange.com/questions/603332/how-can-i-find-the-fifth-root-of-unity