Prove that $3+2\sqrt{3}$ is irrational.

Given :

The given expression is $3+2\sqrt{3}$

To do :

We have to prove that $3+2\sqrt{3}$ is irrational.

Solution :

Let $2 + √6$$3+2\sqrt{3}$ be a rational number

$This\; implies,$

$3+2\sqrt{3}$ = $\frac{a}{b}$, where a and b are integers

This implies,

$2√3 = \frac{a}{b}-3$

$2√3 = \frac{a-3b}{b}$

$√3 = \frac{a-3b}{2b}$

Here, $\frac{a-3b}{2b}$ is a rational number because $a-3b$ and 2b are integers.

This should be a rational number since a, 3b and 2b are all integers.

This implies √3 is a rational number which is a contradiction.

So our assumption that $3+2√3$ is a rational number is incorrect.

Therefore,  $3+2√3$ is an irrational number.

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Updated on: 10-Oct-2022

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