Question

Prove that root 3 + 2 root 5 is an irrational number.​

Answer 1

Answer:

Please see the attachment below

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Assume that 3+ 2 root 5 is rational

Answer 2

To prove =3+2√5 is irrational

proof

$let \: us \: assume \: to \: the \: contrary \: that \: 3 + 2 \sqrt{5} \: is \: rational \\ that \: is \: we \: an \: find \: 3 + 2 \sqrt{5} in \: the \: form \: of \: \frac{p}{q} where \: p \: and \ \\ : q \: are \: integer \: and \: q \: is \: other \: than \: 0 \\ since \: \frac{p}{q} = 3 + 2 \sqrt{5} \\ = > \frac{p}{q} - 3 = 2 \sqrt{5} \\ = > (\frac{p}{q} - 3) \frac{1}{2} = \sqrt{5} \\ we \: conclude \: that \: \sqrt{5} \: is \: rational \\ that \: contradict \: the \: fact \: that \: \sqrt{5} \: is \: irrational \\ \\ so \: our \: assumation \: was \: wrog \: that \: 3 + 2 \sqrt{5} is \: rational \\ \\ therefore\: 3 + 2 \sqrt{5} is \: irrationl$

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