Prove that $5-\sqrt{3}$ is irrational.

 Given :

The given number is $5-\sqrt{3}$.
 

To do :

We have to prove that $5-\sqrt{3}$ is irrational.

Solution :

Let us assume $5-\sqrt{3}$ is rational.

Hence, it can be written in the form of $\frac{a}{b}$, where a, b are co-prime, and b is not equal to 0.

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

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

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

Here, a, b and 5 are integers.

So, $\frac{5b - a}{b} $ is rational.

But, we know that $\sqrt{3}$ is an irrational number.

This contradicts the assumption, $5-\sqrt{3}$ is rational.

Therefore, $5-\sqrt{3}$ is an irrational number.