Question

Show that x+ root y is irrational if x is rational and root y is irrational.

Answer 1

To show: x+√y is irrational.

Let x+√y be a rational number

Therefore,

x+√y= p/q (where p and q are co-primes.)

√y=p-xq/q

Since p,xq and q are integers,p-xq/q is rational

==> √y is also rational

But given √y is irrational! This contradiction arises due to our wrong assumption that x+√y is rational..

==> x+√y is irrational! Hence proved!

Hope it helps :)