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Show that $2+\sqrt{2}$ is not rational.
Given :
The given number is $2+\sqrt{2}$
To do :
We have to prove that $2+\sqrt{2}$ is not rational.
Solution :
Let us assume $2+\sqrt{2}$ 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.
$2+\sqrt{2}=\frac{a}{b}$
$\sqrt{2} = \frac{a}{b} - 2$
$\sqrt{2} = \frac{a -2b}{b}$
Here, a, b and $-2$ are integers.
So, $\frac{a -2b}{b}$ is a rational number.
But, we already know that, $\sqrt{2}$ is an irrational number.
This contradicts the assumption, $2+\sqrt{2}$ is rational.
Therefore, $2+\sqrt{2}$ is not a rational number.
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