Question

please solve this problem​

Answer 1

Answer:

- 24√3

Question: If x = 2 - √3 , find the value of (x - 1/x)³.

Step-by-step explanation:

$\sf{Given, \: x = 2 - √3. \: \: \: So, \: \frac{1}{x} = \frac{1}{2 - √3} }$

$\sf{Rationalizing \: the \: denominator : } \\ \sf{ \implies\frac{1}{x} = \frac{1}{2 - √3} \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } } \\ \\ \sf{ \implies\frac{1}{x} = \frac{2 + \sqrt{3} }{(2 - \sqrt{3} )(2 + \sqrt{3}) } } \\ \\ \sf{ \implies\frac{1}{x} = \frac{2 + \sqrt{3} }{(2 ){}^{2} - (\sqrt{3} ) {}^{2} } } \\ \\ \sf{ \implies \frac{1}{x} = \frac{2 + \sqrt{3} }{4 - 3} } \\ \\ \sf{ \implies\frac{1}{x} = 2 + \sqrt{3} }$

$\sf{Therefore,}$

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

$=> ( (2 - \sqrt3) - (2 + \sqrt3) )^3$

$=> ( - \sqrt3 - \sqrt3 )^3$

$=> ( - 2\sqrt3 )^3$

$=> - 24\sqrt3$

Answer 2

Answer:

Given :

x = 2 -√{3}

To find :

$\red{\boxed{ \bigg(x - \frac{1}{x} \bigg) {}^{3}}}$

Solution :

$\begin{gathered} \sf \: x = 2 - \sqrt{3} \\ \\ \sf \: \frac{1}{x} = \frac{1}{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} } \\ \\ \ \sf \: \frac{1}{x} = \frac{2 + \sqrt{3} }{(2) {}^{2} - ( \sqrt{3} ) {}^{2} } \\ \\ \sf \: \frac{1}{x} = \frac{2 + \sqrt{3} }{4 - 3} \\ \\ \sf \: \frac{1}{x} = 2 + \sqrt{3} \end{gathered}$

Now,

$\begin{gathered} \sf \: x - \frac{1}{x} = 2 - \sqrt{3} - 2 - \sqrt{3} \\ \\ \sf \: x - \frac{1}{x} = - 2\sqrt{3} \end{gathered}$

And,

$\begin{gathered}\bigg(x - \frac{1}{x} \bigg) {}^{3} \\ \\ - \bigg(2 \sqrt{3} \bigg) {}^{3} \\ \\ \therefore \boxed {\bold{ \pink{- 24 \sqrt{3}}}}\end{gathered}$

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