Question

If a is rational and root b is irrational prove that a+root b is irrational

Answer 1

let root(a)+root(b)=r
where r is a rational number 
[root(a)+root(b)]2=r2 
a+b+2root(ab)=r2 
root(ab)=(r2-a-b)/2
(r2-a-b)/2 is a rational no.but root(ab)is irrational
here arised a contradiction due to our wrong assumption that root(a)+root(b) is rational 
hence root(a)+root(b) is irrational

Answer 2

$\textbf{Answer is in Attachment !}$