Prove that


Prove that $(2+\sqrt{3})$ is irrational.


Let $(2+\sqrt{3})$ be rational.

Then, both $(2+\sqrt{3})$ and 2 are rational.

$\therefore\{(2+\sqrt{3})-2\}$ is rational [ $\because$ Difference of two rational is rational]

$\Rightarrow \sqrt{3}$ is rational.

This contradicts the fact that $\sqrt{3}$ is irrational.

The contradiction arises by assuming $(2+\sqrt{3})$ is rational.

Hence, $(2+\sqrt{3})$ is irrational.

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