Question

prove that root x + root Y is irrational where x and y are prime

Answer 1

Answer and Explanation :

To prove : $\sqrt{x}+\sqrt{y}$ is irrational where x and y are prime.

Proof :

We know $\sqrt{x}$ and $\sqrt{y}$ are irrational number.

Let us assume $\sqrt{x}+\sqrt{y}$  are rational number.

So, $\sqrt{x}+\sqrt{y}=\frac{p}{q}$

where p and q are integers and q is non-zero.

$\sqrt{x}=\frac{p}{q}-\sqrt{y}$

Squaring both side,

$(\sqrt{x})^2=(\frac{p}{q}-\sqrt{y})^2$

$x=\frac{p^2}{q^2}+y-\frac{2\sqrt{y}p}{q}$

$\frac{2\sqrt{y}p}{q}=\frac{p^2}{q^2}+y-x$

$2\sqrt{y}p=q(\frac{p^2}{q^2}+y-x)$

$\sqrt{y}=\frac{q(\frac{p^2}{q^2}+y-x)}{2p}$

RHS is a rational number and LHS is an irrational number.

Accordingly Irrational= Rational

Thus it is not possible.

So our assumption is wrong

Hence, $\sqrt{x}+\sqrt{y}$ is a irrational number.