Question

If a =2+/3. than find the vaalue of a+1/a

Answer 1

$\huge\mathfrak{Answer:}$

Given:

• We have been given that a = 2+√3.

To Find:

• We need to find the value of a - 1/a

Solution:

$\sf{As \: it \: is \: given \: that \: a = 2 + \sqrt{3} }$

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

Now, on rationalizing the denominator, we have

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

$\sf{Using \: the \: identity \: : (a + b)(a - b) = {a}^{2} - {b}^{2} }$

$\sf{We \: have}$

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

$\implies\sf{ \dfrac{1}{a} = \dfrac{2 - \sqrt{3} }{4 - 3} }$

$\implies\sf{ \dfrac{1}{a} = 2 - \sqrt{3} }$

$\sf{Now \: we \: need \: to \: find \: the \: value \: of \: a - \dfrac{1}{a}}$

$\sf{We \: have}$

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

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

$\implies\sf{a - \dfrac{1}{a} = \sqrt{3} + \sqrt{3} }$

$\implies\sf{a - \dfrac{1}{a} = \sqrt[2]{3} }$

Hence, the value of a - 1/a is 2√3.