Q. 33.5( 6 Votes )
If p and q are co-prime positive integers then prove that 
is an irrational number.
Answer :
Let √p + √q be a rational number.
Let √p + √q = x where x is integral number.
Squaring both sides.
(√p + √q)2 = x2
p + q + 2√pq = x2
Since p and q are co-prime positive integers. So root of p and root of q will definitely be an irrational numbers as they are not perfect squares. So √pq has to be an irrational number.
So the assumption made at the beginning of the problem is false.
So it is proved that √p + √q is an irrational number.
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