Find the fourth root of of -16i

let Z is the fourth root of  of -16i

  Z= (-16i)^1/4

   ={(16)(-1)(i) }^1/4

    ={  16i³ }^1/4                          (i² =-1)

     =2 {( (i)⁴ )^3/16 }                     (i⁴ =1)

      Z=2

Actually, its complex. And there are four of them. And each one has an imaginary component equal in magnitude to its real components magnitude, because they lie on the y = x and y = -x lines, 45° incline/decline from the x(real)-axis and each root is therefore 90° rotated from the next. 4th root of -16 = ±√2 ± i√2. Anyway, if you take a purely positive real number and find its fourth root, you will get both positive and negative, purely real and purely imaginary roots. If you take a positive or negative, pure imaginary or real number and raised it to the fourth power, it will be a positive real number.

So, the only option for a fourth root of a negative real would be elsewhere on the complex plane. I hope this sheds some light on your question. I know it doesn't answer it directly as you intended it, but maybe helps you somewhat with it!