Let us assume that ` ( sqrt2 +sqrt3)` is irrational.
Then, there exist co-primes a and b such that
` sqrt2 + sqrt3= a/b`
` Rightarrow sqrt3 = a/b -sqrt2`
` Rightarrow (sqrt3)^(2) = ( a/b -sqrt2)^(2)`
` Rightarrow 3 = a^(2)/b^(2) – (2a)/b sqrt2 +2`
` Rightarrow (2a)/b sqrt2 = a^(2)/b^(2) -1`
` Rightarrow sqrt2 = (a^(2) -b^(2))/(2ab)`
Since a and b are intergers, so ` (a^(2) -b^(2))/(2ab)` is rational.
Thus, ` sqrt2` is also rational.
But, this contradicts the fact that ` sqrt2` is irrational , so, our assumption is incorrect.
Hence ` (sqrt2+ sqrt3)` is irrational.