Question

if a=2+root3 find the value a-1/a​

Answer 1

Step-by-step explanation:

Here is the answer you were looking for:

\begin{gathered}a = 2 + \sqrt{3} \\ \\ \frac{1}{a} = \frac{1}{2 + \sqrt{3} } \\ \end{gathered}a=2+3a1=2+31

On rationalizing the denominator we get,

\begin{gathered} \frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \end{gathered}a1=2+31×2−32−3

Using the identity :

(x + y)(x - y) = {x}^{2} - {y}^{2}(x+y)(x−y)=x2−y2

\begin{gathered} \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{gathered}a1=(2)2−(3)22−3a1=4−32−3a1=2−3a−a1

Putting the values,

\begin{gathered}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{gathered}a−a1=(2+3)−(2−3)a−a1=2+3−2+3a−a1=3+3a−a1=23