Answer :

Let √p + √q be a rational number.

Let √p + √q = x where x is integral number.

Squaring both sides.

(√p + √q)^{2} = x^{2}

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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