Question

please solve it .....​

Answer 1

letsqrt(n-1)+sqrt(n+1)be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0squaring on both sides we get n-1+n+1+2sqrt(n^2-1)2n+2sqrt(n^2-1)=p^2/q^22(n+sqrt(n^2-1))=p^2/q^22(n+sqrt(n^2-1))q^2=p^2this mean 2 dividesp^2and also divides p.then let p=2k for any integer kthen 2(n+sqrt(n^2-1))=(2k)^2/q^22(n+sqrt(n^2-1))=4k^2/q^2q^2=2k^2/(n+sqrt(n^2-1))so 2 dividesq^2and also qp and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. sosqrt(n-1)+sqrt(n+1)is irrational.