Question

giving the root 2 is irrational proof that 5 + 3 root 2 is irrational number

Answer 1

Hey there!
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Prove
$5 + 3 \sqrt{2} \: is \: irrational$

$let \: us \: assume \: 5 + 3 \sqrt{2} \: is \: rational$

$5 + 3 \sqrt{2} = \frac{a}{b} \:$
$where \: a \: and \: b \: are \: integers \: and \: b \: is \: not \: equal \: to \: 0$

$3 \sqrt{2} = \frac{a}{b} - 5$

$= \frac{a - 5b}{b}$

$\sqrt{2} = \frac{a - 5b}{3b}$


$\frac{a - 5b}{3b} \: is \: rational$


(as a and b are integers)


$but \: \sqrt{2} \: is \: irrational$

So, This contradicts the assumption that


$5 + 3 \sqrt{2} \: is \: rational$

$hence \: 5 + 3 \sqrt{2} \: is \: irrational$

Hence, the proof.

Hope helped.