Question

prove that √7+√6is an irrational, where p, q are primes​

Answer 1

Solution:-

=> Let us assume that ( √7 +√6 ) is irrational

=> Thus , there exists co - primes a and b such that

=> √7+√6 = a/b

=> √6 = a/b - √7 [ Squaring on both side ]

We get

=> ( √6 )² = ( a/b - √7 )²

=> 6 = a²/b² - (2a/b)× √7 + 7

=> (2a/b)×√7 = a²/b² - 1

=> √7 = (a² - b² )/2ab

Since a and b are integers so (a² - b² )/2ab is rational

thus √7 is also rational

But this contradicts the fact that √7 is irrational. So , our assumption is incorrect

Hence ( √7 + √6 ) is irrational Number

Hence proved