Question

Prove (5+3‚àö2) irrational where ‚àö2 is irrational

Answer 1

Hey friend, Harish here.

Here is your answer:

Given that,

√2 is an irrational number.

To  prove,

5 + 3√2 is an irrational number.

Assumption:

Let 5 + 3√2 be a rational  number.

Proof:

As 5 + 3√2 is assumed to be rational , then it must be of the form p/q, Where q≠0.

Then,

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

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

⇒ $\sqrt{2}= \frac{p-5q}{3q}$

We know that,

$\sqrt{2}\ is \ irrational\ (Given)$

$\frac{p-5q}{3q} \ is \ rational$

As rational ≠ irrational.

We contradict the statement that 5+3√2 is rational.

Therefore it is irrational.
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Hope my answer is helpful to you.