Wednesday, February 15, 2017

elementary number theory - Is fracsqrt7sqrt[3]15 rational or irrational?



Is 7315 rational or irrational? Prove it.




I am having a hard time with this question. So far what I did was say, assume it's rational, then 7315=xy7y=x315


I then showed the product of a rational number and an irrational number is irrational so the expression above is irrational on the left and right side. I can't get to a contradiction to prove it's irrational so I'm currently thinking it is rational but I don't know how to go about proving it.


Answer



Factoring 73y6=152x6 into primes, 7 has odd power 3+6j on LHS, but even 6k on RHS


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