Question

$3 + \sqrt{5 \:} \: prove \: it \: is \: a \: irrational \: number$

Answer 1

$let \: 3 + \sqrt{5} \: be \: rational$

$3 + \sqrt{5} = \frac{x}{y} (x \: and \: y \: are \: integers)$

$\sqrt{5} = \frac{x}{y} - 3$

Since a and b are integers so x/y-3 is also rational.This shows that root 5 is also rational but this contradicts the fact that root 5 is irrational.

$so \: 3 + \sqrt{5} \: is \:irrational$

Answer 2

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