Question

If a= 2+ √3 , then find the value of a - 1/ a.​

Answer 1

GIVEN :-

$\\ \sf \: a = 2 + \sqrt{3}$

TO FIND :-

$\\ \sf \: a - \dfrac{1}{a}$

SOLUTION :-

We have ,

$\\ \sf \: a = 2 + \sqrt{3}$

Taking reciprocal,

$\\ \sf \: \dfrac{1}{a} = \dfrac{1}{2 + \sqrt{3} }$

Rationalising the denominator,

Rationalising factor is 2 - √3.

Multiplying numerator and denominator with (2 - √3).

$\\ \implies\sf \: \dfrac{1}{a} = \dfrac{1}{2 + \sqrt{3} } \times \dfrac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \\ \\ \implies \sf \: \dfrac{1}{a} = \dfrac{2 - \sqrt{3} }{(2 + \sqrt{3} )(2 - \sqrt{3} )}$

In denominator,

★ (a+b)(a-b) = a² - b²

• a = 2
• b = √3

Putting values,

$\\ \implies\sf \: \dfrac{1}{a} = \frac{2 - \sqrt{3} }{ {2}^{2} - ( { \sqrt{3} )}^{2} } \\ \\ \\ \implies \sf \: \dfrac{1}{a} = \dfrac{2 - \sqrt{3} }{4 - 3} \\ \\ \\ \implies \underbrace{\sf \: \dfrac{1}{a} = 2 - \sqrt{3}}$

We have ,

• a = 2 + √3
• 1/a = 2 - √3

$\\ \implies\sf \: a - \dfrac{1}{a} = 2 + \sqrt{3} - (2 - \sqrt{3} ) \\ \\ \\ \implies \sf \: a - \dfrac{1}{a} = 2 + \sqrt{3} - 2 + \sqrt{3} \\ \\ \\ \implies \underline{\overline{\boxed{\sf \: a - \dfrac{1}{a} = 2 \sqrt{3} } }}$