If root xy is an irrational number then prove that root x +root y is irrational - Maths - Real Numbers

Question

If root xy is an irrational number then prove that root x +root
y is irrational

Shridhar Kaushik · Accepted Answer

Given $\sqrt{x y}$ is an irrational number To prove : $\sqrt{x} + \sqrt{y}$ is irrational Let if possible $(\sqrt{x} + \sqrt{y})$ be rational $\Rightarrow (\sqrt{x} + \sqrt{y}) = \frac{p}{q} for q \neq 0 \Rightarrow (\sqrt{x} + \sqrt{y})^{2} = \frac{p^{2}}{q^{2}} \Rightarrow x + y + 2 \sqrt{x y} = \frac{p^{2}}{q^{2}} \Rightarrow 2 \sqrt{x y} = \frac{p^{2}}{q^{2}} - (x + y) (1)$ Since $\sqrt{x} + \sqrt{y}$ is rational â‡’ x and y will be rational âˆ´ R H S of equation (1) is rational But L H S of (1) is given to be irrational â‡’ We arrive to a contradiction Hence our supposition that $\sqrt{x} + \sqrt{y}$ is rational is wrong âˆ´ $\sqrt{x} + \sqrt{y}$ is irrational