Prove that


Prove that $\frac{3}{\sqrt{5}}$ is irrational, given that $\sqrt{5}$ is irrational.



Let us assume that $\frac{3}{\sqrt{5}}$ is a rational number.

Thus, $\frac{3}{\sqrt{5}}$ can be represented in the form of $\frac{p}{q}$, where $p$ and $q$ are integers, $q \neq 0, p$ and $q$ are co-prime numbers.


$\Rightarrow p \sqrt{5}=3 q$

$\Rightarrow \sqrt{5}=\frac{3 q}{p}$

Since, $\frac{3 q}{p}$ is rational $\Rightarrow \sqrt{5}$ is rational

But, it is given that $\sqrt{5}$ is an irrational number.

Therefore, our assumption is wrong.

Hence, $\frac{3}{\sqrt{5}}$ is an irrational number.


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