Question

prove that 2+5 √3 is an irrational number​

Answer 1

$let \: \: 2 + 5 \sqrt{3} \: \: is \:rational \\ 2 + 5 \sqrt{3} = \frac{a}{b} (where \: a \: and \: b \: are \: integers \: and \: bis \: not \: equal \: to \: zero) \\ 5 \sqrt{3} = \frac{a}{b} - 2 \\ 5 \sqrt{3} = \frac{a - 2b}{b} \\ \sqrt{3} = \frac{a - 2b}{5b} \\ as \: we \: know \: that \: \frac{a - 2b}{5b} is \: a \: rational \: number \\ so \: \sqrt{3} will \: be \: a \: rational \: number \\ but \: \sqrt{3} is \: irrational. \\ \: this \: contradicts \: our \: assumption \: that \: 2 + 5 \sqrt{3} is \: rational \\ therefore \: 2 + 5 \sqrt{3} is \: irrational$