Question

if x=1/2-√3 find (x+1/x)^3​

Answer 1

Question :

If $x = \dfrac{1}{2 - \sqrt{3} }$

Find $(x + \frac{1}{x}) {}^{3}$

Solution :

We have to find $(x + \frac{1}{x}) {}^{3}$

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Given :$x = \frac{1}{2 - \sqrt{3} }$

Rationalise the denominator

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

$x = \dfrac{2 + \sqrt{3} }{4 - 3}$

$x = 2 + \sqrt{3}$

Now ;

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

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Now;$(x + \frac{1}{x}) {}^{3}$

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

$= (4) {}^{3}$

$= 64$

which is the required solution!