Prove that $\frac{7\sqrt{3}}{2}$ is an irrational number.

Given :

The given number is $\frac{7\sqrt{3}}{2}$.

To do :

We have to prove $\frac{7\sqrt{3}}{2}$ is an irrational number.

Solution : 

Let us assume $\frac{7\sqrt{3}}{2}$ 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.

$\frac{7\sqrt{3}}{2}=\frac{a}{b}$

$ \sqrt{3} = \frac{2a}{7b}$

Here, a and b are integers.

So, $\frac{b}{a}$ is a rational number. Which means 

We aready know that $\sqrt{3}$ is an irrational number.

This contradicts the assumption, $\frac{7\sqrt{3}}{2}$ is rational.

Therefore, $\frac{7\sqrt{3}}{2}$ is an irrational number.

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Updated on: 10-Oct-2022

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