perplexus.info :: Just Math : Rational number's conversion

Rational number's conversion (Posted on 2017-06-18)

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

(a²+b³)/ (c⁵+d⁷)

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

No Solution Yet Submitted by Ady TZIDON

Rating: 5.0000 (1 votes)

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Answer | Comment 4 of 5 |

Let the rational number be x/y. Let a=x^3*y^2, b=x^5*y^2, c=x*y, and 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/y.

Posted by Math Man on 2017-06-20 07:36:16

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