1.

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

Answer»

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!


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