Question

If x = 2 –√3, find the value of
$(x - \frac{1}{x} ) {}^{3}$
​

Answer 1

Answer:

Value of ( x - 1/x )³ is - 24√3.

Step-by-step explanation:

x = 2 - √3

Find the value of 1/x

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

Multiply both numerator and denominator with 2 + √3

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

$\implies \dfrac{1}{x} = \dfrac{2 + \sqrt{3} }{(2 - \sqrt{3})(2 + \sqrt{3}) }$

Using algebraic identity (x - y) (x + y) = x² - y²

$\implies \dfrac{1}{x} = \dfrac{2 + \sqrt{3} }{ {2}^{2} - {( \sqrt{3}) }^{2} }$

$\implies \dfrac{1}{x} = \dfrac{2 + \sqrt{3} }{ 4- 3 }$

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

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

Now ( x - 1/x )³

$\implies \bigg(x - \dfrac{1}{x} \bigg)^3= \bigg(2 - \sqrt{3} -(2 + \sqrt{3} ) \bigg)^3$

$\implies \bigg(x - \dfrac{1}{x} \bigg)^3= \big(2 - \sqrt{3} -2 - \sqrt{3} \big)^3$

$\implies \bigg(x - \dfrac{1}{x} \bigg)^3= \big( - 2 \sqrt{3} \big)^3$

$\implies \bigg(x - \dfrac{1}{x} \bigg)^3= ( - 2 )^3( \sqrt{3})^3$

$\implies \bigg(x - \dfrac{1}{x} \bigg)^3= - 8( 3\sqrt{3})$

$\implies \boxed{ \bigg(x - \dfrac{1}{x} \bigg)^3= - 24\sqrt{3}}$

Therefore the value of ( x - 1/x )³ is - 24√3.