Question

Given that under root 2 is rational prove that the bracket mein 5 + 3 under root 2 bracket close is an irrational number

Answer 1

$let(5 + 3 \sqrt{2} )is \: rational \: number \\ then \: 5 + 3 \sqrt{2} = \frac{p}{q} it \: is \: simplest \: \\ form \: i.e \: p \: and \: q \: no \: common \: \\ factor \: other \: than1 \\ 3 \sqrt{2} = \frac{p}{q} - 5........rational \: number \\ \sqrt{2} = \frac{1}{3} \times ( \frac{p}{q} - 5).....rationl \: number. \\ but \: \sqrt{2 } is \: irrationl \: given\\ whic \: contradict \: our \: assumption \\ so \: 5 + 3 \sqrt{2} is \: irrtional \: number \\ |note| \\ (+ - \times \div )of \: two \: rational \: number \\ \: is \: also \: rational$