Question

prove that root p+root q is an irrational ,where p,q are primes​

Answer 1

$let \sqrt{p} + \sqrt{q} = \frac{a}{b} where \: a \: and \: b \: \\ have \: no \: common \: factor \: other \\ \: than \: 1 \\ squaring \: both \: side \\ p + q + 2 \sqrt{pq} = \frac{ {a}^{2} }{ {b}^{2} } (rational) \\ \sqrt{pq} = (\frac{ {a}^{2} }{ {b}^{2} } - p - q) \times \frac{1}{2} (rational \: number) \\ whic \: are \: contradict \\ so \: our \: assumpation \: is \: wrong \\ and \sqrt{p} + \sqrt{q} \: are \: irrtional \: number$