Question

if a=2+√3 , find value of a-1/a​

Answer 1

Answer:

Heya friend,

Here is the answer you were looking for:

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

On rationalizing the denominator we get,

$$\begin{lgathered}\frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\\end{lgathered}$$

Using the identity :

$$(x + y)(x - y) = {x}^{2} - {y}^{2}$$

$$\begin{lgathered}\frac{1}{a} = \frac{2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{a} = 2 - \sqrt{3} \\ \\ a - \frac{1}{a}\end{lgathered}$$

Putting the values,

$$\begin{lgathered}a - \frac{1}{a} = (2 + \sqrt{3} ) - (2 - \sqrt{3} ) \\ \\ a - \frac{1}{a} = 2 + \sqrt{3} - 2 + \sqrt{3} \\ \\ a - \frac{1}{a} = \sqrt{3} + \sqrt{3} \\ \\ a - \frac{1}{a} = 2 \sqrt{3}\end{lgathered}$$