There are already two puzzles on this site asking to find three real roots to an equation of the form:
(x^3+k*x)^(1/5) = (x^5-k*x)^(1/3)

The two puzzles are k=2 here and k=20 here.

I challenge you to find a rational value of k such that there are four nonzero rational roots to the equation above.

(In reply to Heuristic approach - spoiler by Harry)

Excellent!  Not just one rational k, but an infinite number of them!

Posted by Steve Herman on 2016-03-28 07:27:33