Question

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

Answer 1

Here is the answer you were looking for:
a = 2 +  \sqrt{3}  \\  \\  \frac{1}{a}  =  \frac{1}{2 +  \sqrt{3} }  \\  
On rationalizing the denominator we get,
\frac{1}{a}  =  \frac{1}{2 +  \sqrt{3}  }   \times  \frac{2 -  \sqrt{3} }{2 -  \sqrt{3} }  \\  
Using the identity :
(x + y)(x - y) =  {x}^{2}  -  {y}^{2}  
\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}  
Putting the values,
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}  

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