Question

Prove that 8+5√2 is an irrational.

Answer 1

Answer:

$let \: 8 + 5 \sqrt{2 \: } \: is \: a \: rational \: number \: \\ so \: 8 + 5 \sqrt{2} = p \q \: where \: p \: and \: q \: both \: are \: integers \: and \: q \: not \: equal \: to \: 0 \: \\ now \: 8 + 5 \sqrt{2} = p \q \\ 5 \sqrt{2} = p \q - 8 \\ 5 \sqrt{2} = p - 8q \5q \\ \sqrt{2} \: \: p - 8q \5q \\ p \: and \: q \: are \: both \: integers \: so \: p \: - 8q \: \5q \: is \: a \: rational \: number \: \\ but \: \sqrt{2} \: is \: an \: irrational \: number \: \\ so \: rational \: number \: cannot \: be \: equal \: to \: irrational \: number \: therefore \: 8 + 5 \sqrt{2} \: is \: irrational \: number \:$

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