All of the derived eq's from the fourth root, for anyone curious. These are included for all the previous roots in some way, so I thought including this too would be interesting. Graphs were made by substituting my real eq into every possible y value in my eq for bi.
I also needed to say somewhere that- for the second root, b is accurate for all values, the third is accurate for b for all values as well, both are made from curves that are related back to a standard complex number, (the third/second degree graph) and can therefor be fully detailed on a 2d plain. The fourth root is only accurate for b up to 270 degrees (I have it in radians in the graph) or else it is only roughly approximate (same when a is less than 1), but the entire graph rotates when a is negative, rendering all curves inaccurate, and barely approximate at best. The fifth complex root is not related to the real fifth root, and the equation becomes inaccurate at a>5, but can only rotate up to .5 degrees before no longer being 1:1 with the real plain, and this is true even if I use an equation that works as long as 1<a<5. Confusing? Yeah, but if you look a bit closer and research a bit, there are a few reasons for this, namely, the fifth root is infinately differentiable with my method, never simplifies, and the fourth root is only differentiable as long as the real part is positive.
