Prove that (2 + √3) / 5 is an irrational number

Q) Prove that (2 + √3) / 5 is an irrational number. It is given that √3 is an irrational number.

Ans:

Let’s start by considering $\frac{(2 + \sqrt 3)}{5}$ is a rational number.

∴ $\frac{(2 + \sqrt 3)}{5} = \frac{p}{q}$ (here p and q are integers and q ≠ 0)

∴ $(2 + \sqrt 3) = \frac{5 p}{q}$

∴ $\sqrt 3 = \frac{5 p}{q} - 2$

∴ $\sqrt 3 = \frac{5 p - 2 q}{q}$

Since p and q are integers, so, $\frac{5 p - 2 q}{q}$ is a rational number.

If RHS is a rational number, then LHS will also be a rational

Therefore √3 is a rational number.

But it contradicts the given condition (∵ given that √3 is an irrational number)

Therefore, it is confirmed that $\frac{(2 + \sqrt 3)}{5}$ is an irrational number.

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