Question

answer for the question given in the picture​

Answer 1

Question :

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

Solution :

Given that

a = 2 + √3

Here,

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

Now,

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

Answer 2

$\large \green{ \bf Given : }$

$a = 2 + \sqrt{3}$

$\bf \large \blue {To \: Find : }$

$a - \frac{1}{a}$

$\large \red{ \bf Solution : }$

$a = 2 + \sqrt{3} \\ \\ \frac{1}{a} = \frac{1}{2 + \sqrt{3} }$

By rationalisation

$\frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3}} \\ \\ \frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{a} = 2 - \sqrt{3}$

$a - \frac{1}{a} \\ \\ \Longrightarrow 2 + \sqrt{3} -( 2 - \sqrt{3}) \\ \\ \large{ \boxed{ \bf \pink{ \Longrightarrow 2\sqrt{2}}} }$