Show that every positive rational number can be presented as a ratio of powers' sum in the following way:

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(a**^{2}+b^{3})/ (c^{5}+d^{7})

where a,b,c,d are positive integers, not necessarily distinct.

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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).