Question

prove that root 2 + 3 by root 2 is an irrational number

Answer 1

If possible let root2 + 3/root2 be an rational number.
Then root2 + 3/root2 = p/q where p and q are co primes.
By taking LCM,
(root2 × root2 + 3)/root2 = p/q
(2+3)/root2 = p/q
5/root2 = p/q
root2 = 5q/p .....(1)
We know that root 2 is irrational but 5q/p is rational.
But this it contradictory to the above equation (1).
This contradiction arose because of our incorrect assumption of root2 + 3/root2 to be rational.
This proves that root2 + 3/root2 is irrational.

Hope it helps!