Question

prove that the following are irrational 3+5root2​

Answer 1

PROOF

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let suppose to the contrary ,3+5√2 is rational.

$that \: \: is \: we \: can \: find \: a \: and \: b(other \: than \: 0) \\ such \: that \: 3 + 5 \sqrt{2} = \frac{a}{b} \\ \\ or \: 5 \sqrt{2} = \frac{a}{b} - 3 \\ \\ or \: \sqrt{2} = \frac{1}{5} ( \frac{a}{b} - 3).....(i) \\ \: this \: is \: denied \: that \: fact \sqrt{2} is \: irrational \: as \: rhs \: in \: \\ equation \: (i) \: is \: rational \\ \\ since \: our \: assumation \: is \: wrong \\ therefore \: 3 + 5 \sqrt{2} \: is \: irrational$

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