Question 14
Prove that √p+√q is irrational, where p and q are primes.
Let us suppose that √p+√q is rational.
Again, let √p+√q=a, where a is rational.
Therefore, √q=a−√p
On squaring both sides, we get
q= a2+p−2a√p [∵ (a−b)2=a2+b2−2ab]
Therefore, √p=a2+p−q2a.
This contradicts our assumption, as the right-hand side is a rational number whereas √p is irrational.
Hence, √p+√q is irrational.