Question

if x is rational and root Y is irrational then prove that X + root y is irrational

Answer 1

Let us assume to the contrary that x+√y is rational
So x+√y can be written in the form a/b,where a and b are co-primes and b not equal to 0
x+√y=a/b
√y=a/b-x
√y=a-bx/b
Since x,a,b are all integers, therefore they are rational
But this contradicts the fact that√y is irrational
Hence our assumption is incorrect.
Therefore x+√y is irrational.

Answer 2

Answer:

Let us assume to the contrary that x+√y is rational

So x+√y can be written in the form a/b,where a and b are co-primes and b not equal to 0

x+√y=a/b

√y=a/b-x

√y=a-bx/b

Since x,a,b are all integers, therefore they are rational

But this contradicts the fact that√y is irrational

Hence our assumption is incorrect.

Therefore x+√y is irrational.