Question

Given that 2 is irrational number, prove that ( 5+3 2 ) is an irrational number.

Answer 1

Answer:

Assume that 5 + 3√2 be rational

$5 + 3 \sqrt{2} = \frac{p}{q} \: \: where \: \: p \: \: and \: \: q \: \: are \: \: integers$

$5 + 3 \sqrt{2} = \frac{p}{q}$

$3 \sqrt{2} = \frac{p}{q} - 5$

$3 \sqrt{2} = \: \frac{pq - 5}{q}$

$\sqrt{2} = \frac{pq - 5}{3q}$

LHS is irrational

RHS is rational.

Irrational cannot be equal to rational

Therefore our assumption is wrong,

Hence, 5 + 3√2 is irrational.

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