Question

can you please solve my problem friend's
$prove \: that \: 3 + 2 \sqrt{3 \:} is \: an \: \: irrantional \: number \:$
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Answer 1

$\large{\underline{\boxed{ Answer}}}$

Let $3 + 2 \sqrt{3}$

be a rational number then, it can be expressed in the form of P/q where p and q are co - prime number and q ≠ 0.

Now,

$\implies$ $3 + 2 \sqrt{3} = \frac{p}{q}$

$\implies$ $2 \sqrt{3} = \frac{p}{q} - 3$

$\implies$ $\sqrt{3} = \frac{p}{2q} - 3$

we know that every operation of rational number is always a rational number .

Therefore, division of rational is also a rational number.

using this fact,

we know that √3 is an irrational number,

L. H. S is an irrational number but R. H. S is a rational number.

hence, our supposition is wrong$3 + 2 \sqrt{3}$

is an irrational number.