Question

if x is a rational number and y is irattional number then show that (x+root y) is irrational​

Answer 1

Answer:

since x= rational and y= irrational

let us assume that X+$\sqrt{y}$ iis rational ,then we can find two integers a and b such that x+$\sqrt{y}$ = a/b ,where a and b are co primes and b$\neq 0$  

x+$\sqrt{y}$=a/b

$\sqrt{y}$ =a/b-x

$\sqrt{y}$=a-bx/b

a,b and 1 are integers

∴a-bx/x is rational

∴$\sqrt{y}$ is rational

But this contradicts the fact that √y is irrational

Hence our assumption is incorrect.

Therefore x+√y is irrational.