Answers

2016-03-31T08:46:49+05:30
We know that sqrt(n) is rational if and only if n is a perfect square. 

We assume X = sqrt(n-1) + sqrt(n+1) is rational. Hence X^2 is also rational. X^2 = n - 1 + n + 1 - 2sqrt(n^2 - 1). Therefore sqrt(n^2 - 1) must also be rational. But that means n^2 - 1 must be a perfect square. n^2 is already a perfect square, so this is clearly impossible. 

You can also show that n > sqrt(n^2 - 1) > n - 1, meaning no sqrt(n^2 - 1) is not an integer, hence n^2 - 1 is not perfect square.
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