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.
Building on the two previous posts I finally found a solution.
Let our rational number be expressed as x/y, with x and y integers.
Then make a=x^3*y^2, b=x^5*y^2, c=x*y, d=x^2*y.
Then (a^2+b^3) / (c^5+d^7)
= ((x^3*y^2)^2+(x^5*y^2)^3) / ((x*y)^5+(x^2*y)^7)
= (x^6*y^4+x^15*y^6) / (x^5*y^5+x^14*y^7)
= (x*(x^5*y^4+x^14*y^6)) / (y*(x^5*y^4+x^14*y^6))
Another similar solution can be found with a=x^4*y^3, b=x^2*y^3, c=x*y^2, d=x*y.