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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.
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