(Q5).Prove that √2 + √3 is irrational

√2 + √3

Let us suppose that √2 + √3 is rational.

Let √2 + √3 =
a
b
 where a,b are integers and (b ≠ 0).

Therefore √2 = a/b - √3

2 =
a2
b2
 + 3 - 2
a
b
 √3

Rearranging

2a
b
 √3 =
a2
b2
 + 3 - 2
=
a2
b2
 +
√3 =
a2 + b2
2ab
 +
since a,b are integers,
a2 + b2
2ab
 is rational, so √3 is .

This contradicts the fact that √3 is .

Hence √2 + √3 is

So √2 + √3 is