Question

given that under root 2 is irrational, prove that (5+3 under root 2 ) is an irational number

Answer 1

Heya

Given : √2 is irrational

Let (5+3√2) be a rational number

(5+3√2) = p/q where p,q are integers and q isn't equal to 0

3√2=(p-5q)/q

√2 = (p-5q)/3q

p, q are integers, so (p-5q)/3q is a rational number

As LHS= RHS, √2 becomes rational

This contradicts the fact that √2 is irrational.(given)

This contradiction arose because of our wrong assumption.

Therefore, (5+3√2) is irrational

Hope it helps!

Answer 2

Hey !!

Here is your answer...

Given that =
$\sqrt{2} \\ \\ to \: prove \: = 5 + 3 \sqrt{2} \\ \\ proof \: = let \: \: suppose \: \: 5 + 3 \sqrt{2} = m \: \: is \: \: rational \: \: number. \\ \\ 5 + 3 \sqrt{2} = m \\ \\ 3 \sqrt{2} = m - 5 \\ \\ \sqrt{2} = \frac{m - 5}{3} \\ \\ but \: \: given \: \: that \: \: \sqrt{2} \: \: is \: \: irrational \: \: no. \: \: \\ \\ so \: \: 5 + 3 \sqrt{2} is \: \: \: also \: \: irrational \: \: no. \\ \\ hence \: \: our \: \: assumption \: \: was \: \: wrong \: \:$
THANKS

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