Question

and plz also give me ans of

Answer 1

Given x = 1/2+root 3.

On rationalizing the denominator, we get

$x = \frac{1}{2 + \sqrt{3} } * \frac{1 - 2 \sqrt{3} }{1 - 2 \sqrt{3} }$

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

          $= \frac{2 - \sqrt{3} }{4 - 3}$

          $2 - \sqrt{3}$.



Now,

$(1) x - \frac{1}{x} = 2 - \sqrt{3} - (2 - \sqrt{3})$
    
           $= 2\sqrt{3}$



Now,

$(x - \frac{1}{x})^2 = (2 \sqrt{3})^2$

                                  $= 12$



Now,

On cubing both sides, we get

$(x - \frac{1}{x} )^3 = (2 \sqrt{3})^3$

$x^3 - \frac{1}{x^3} - 3 * x * \frac{1}{x} = 24 \sqrt{3}$

$x^3 - \frac{1}{x^3} = 24 \sqrt{3} + 3$


Hope this helps!