Question

Prove that $\sqrt{2}+\sqrt{3}$ is irrational. [3 MARKS]

Answer 1

Concept : 1 Mark
Application : 2 Marks

Let $\sqrt{2}+\sqrt{3}$ be rational number say '$x$'

$\Rightarrow x=\sqrt{2}+\sqrt{3}$

$x^2=2+3+2\sqrt{6}$

$\Rightarrow x^2=5+2\sqrt{6}$

$\Rightarrow \sqrt{6}=\frac{x^2-5}{2}$

As $x$, 5 and 2 are rationals,

$\Rightarrow \frac{x^2-5}{2}$ is a rational number.

$\Rightarrow \sqrt{6}=\frac{x^2-5}{2}$ is a rational number

which is contradiction to the fact that $\sqrt{6}$ is an irrational number.

Hence our assumption is wrong.

$\Rightarrow \sqrt{2}+\sqrt{3}$ is an irrational number.