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Prove that $\frac{1}{\sqrt{5}}$ is irrational.
Given :
The given number is $\frac{1}{\sqrt{5}}$
To do :
We have to prove that $\frac{1}{\sqrt{5}}$ is irrational.
Solution :
Let us assume $\frac{1}{\sqrt{5}}$ 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.
$\frac{1}{\sqrt{5}}=\frac{a}{b}$
$\frac{b}{a} = \sqrt{5}$
Here, a and b are integers.
So, $\frac{b}{a}$ is a rational number.
We know that $\sqrt{5}$ is an irrational number.
This contradicts the assumption, $\frac{1}{\sqrt{5}}$ is rational.
Therefore, $\frac{1}{\sqrt{5}}$ is an irrational number.
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