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Answer» Let√n−1+√n+1be 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+2√n2−12n+2√n2−1=p2q2 2(n+√n2−1)=p2q2 2(n+√n2−1)q2=p2 this mean 2 dividesp2and also divides p.then let p=2k for any integer k then 2(n+√n2−1)=(2k)2q2 2(n+√n2−1)=4k2q2 q2=2k2/(n+√n2−1) so 2 dividesq2and also q p and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. so√n−1+√n+1is irrational. |
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