Let us assume that` sqrtp + sqrtq` is rational.
Then , there exist co-primes a and b such that
` sqrtp + sqrtq = a/b`
` Rightarrow sqrtp = a/b -sqrtq`
` Rightarrow (sqrtp)^(2) = (a/b -sqrtq)^(2)`
` Rightarrow p = a^(2)/b^(2) – (2a)/bsqrtq + q`
` Rightarrow (2a)/(b) sqrtq = a^(2)/b^(2) + q -p`
` Rightarrow sqrtw = (a^(2) +b^(2)q – b^(2)p) xx b/(2a) = (a^(2)b+b^(3)(q-p))/(2a)`
Since, a,b,p,q are intergers so `(a^(2)b+b^(3)(q-p))/(2a)` is rational
Thus, ` sqrtq` is also rational.
But, q being prime `sqrtq` is irrational.
Since, a contradicition arises so our assumption is incorrect.
Hence, `(sqrtp +sqrtq)` is irrational.