Question

prove √2+√3 is irrational pls I want each and every line

Answer 1

Assume it's rational. Then there are integers m,n such that 
m/n = √2 + √3 
Square both sides: 
m^2 / n^2 = 5 + 2√6 
"Solve" for √6: 
√6 = (m^2 - 5n^2) / (2n^2) 
so if √2 + √3 is rational, then so is √6. 

Let a and b be the integers with gcd(a,b) = 1 such that 
a/b = √6 
Square both sides and multiply by b^2: 
a^2 = 6b^2 
Now, the right side is divisible by 2, so a^2 is divisible by 2, which then implies that a is divisible by 2 (since 2 is prime). Therefore we can write a=2k for some integer k: 
4k^2 = (2k)^2 = 6b^2 
Divide by 2: 
2k^2 = 3b^2 
Now the left side is divisible by 2, so 3b^2 is divisible by 2, from which it follows that b is divisible by 2. 
However, this would mean that 2 divides gcd(a,b) = 1. Contradiction. 

Answer 2

This is the right answer