Show that every positive rational number can be presented as a ratio of powers' sum in the following way:
where a,b,c,d are positive integers, not necessarily distinct.
It turns we already had this problem with Dangerous Expression for Rationals
but nobody solved it.
After reviewing those comments and mulling it over I had an inspiration. x^6 is a power of 3 and is one degree higher than x^5. Similarly, x^8 is a power of 2 and is one degree higher than x^7.
Let a=x^4, b=x^2, c=x, d=x. Then (a^2+b^3)/(c^5+d^7) = (x^8+x^6)/(x^5+x^7) = x*(x^7+x^5)/(x^5+x^7) = x. This works for any nonzero integer x, but does not satisfy the non-integer rationals (x needs to be an integer for a,b,c,d to be integers as stated).