Question

Show that the reciprocal of 2 + root 3 is irrational

Answer 1

$\huge \bf \orange{Hey \: there !! }$


▶ Question :-

→ Show that the reciprocal of 2 + √3 is an irrational number .


$\huge \boxed{\bf \pink{ \mid \underline{ \overline{Solution :- }} \mid}}$


Reciprocal of $2 + \sqrt{3} = \frac{1}{2 + \sqrt{3} } .$

Now, rationalize the denominator.

$= \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} }. \\ \\ = \frac{2 - \sqrt{3} }{ {(2)}^{2} - { (\sqrt{3} )}^{2} } . \\ \\ = \frac{2 - \sqrt{3} }{4 - 3} . \\ \\ = 2 - \sqrt{3} .$


If possible, let ( 2 - √3 ) be rational number. Then,

( 2 - √3 ) is rational number, 2 is rational number.

=> {( 2 - √3 ) - 2 } is rational .

[ °•° Difference of rationals is rational ]

=> - √3 .

This contradicts the fact that - √3 is irrational number .

The contradiction arises by assuming that ( 2 - √3 ) is rational number is wrong .

Hence, ( 2 - √3 ) is an irrational number.


✔✔ Hence, it is proved ✅✅.



THANKS



#BeBrainly.

Answer 2

$\underline{ \huge \bold { solution : }}$

▪️Question : Show that reciprocal of 2 + √3 is an irrational.

_______________________

Given : Reciprocal of 2 + √3 i.e., $\sf \frac{1}{2 + \sqrt{3}}$

To prove : Reciprocal of 2 + √3 is an irrational number.

Proof :

First of all, rationalise the denominator of the reciprocal of 2 + √3.

$\sf \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \sf \frac{2 - \sqrt{3} }{(2) {}^{2} - ( \sqrt{3}) {}^{2} } \\ \\ \sf \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \bf 2 - \sqrt{3}$

After rationalising its denominator, we get (2 - √3) as a result.

Now, let us assume that ( 2 - √3 ) is an irrational number. So, taking a rational number i.e., 2 and subtracting from it.

We have ;

[ 2 - √3 - 2 ]

⇒ - √3

As a result, we get ( - √3 ) which is an irrational number.

Hence, the reciprocal of ( 2 + √3 ) is an irrational number.