Question

if a =2+√3 them find the value of a-1/a​

Answer 1

Given :

• a =2+√3

To find :

• a-1/a

Solution :

⟹$a = 2 + \sqrt{3}$

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

Now we will rationalise the denominator :-

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

we know that :- (a-b)(a+b) = a²-b²

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

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

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

⟹$\dfrac{1}{a} = 2 - \sqrt{3}$

Now we have to find the value of :- a - 1/a

⟹$a - \frac{1}{a} = 2 + \sqrt{3} -( 2 - \sqrt{3})$

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

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

⟹$a - \frac{1}{a} = 0 + 2 \sqrt{3}$

⟹$a - \frac{1}{a} = 2 \sqrt{3}$

Answer : 2√3