Question

show thatroot 5 minus 2 root 3 is an irrational number

Answer 1

If possible let us assume 5-2√3 is rational

So it can be written as 5-2√3 = p/q(where p and q are co prime and q ≠ 0

5-p/q = 2√3
(5q-p)/q = 2√3
√3 = (5q-p)/2q

AS p/q is rational so (5p-1)/2q is also rational but √3 is irrational. This contradiction has arised due to our wrong assumption that 5-2√3 is rational.

So 5-2√3 is irrational