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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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