Question

if x=2-√3 ,then find value of ( x-1/x)^3

Answer 1

Answer:

$\large{\underline{\boxed{\sf - 24 \sqrt {3}}}}$

Step-by-step explanation:

Given that ;

x = 2 - √3

Now, find the value of 1/x.

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

Find the value of x - (1/x).

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

Therefore,

$\implies \sf (x - \frac{1}{x} ) {}^{3} = ( - 2 \sqrt{3}) {}^{3} \\ \\ \implies \sf (x - \frac{1}{x} ) {}^{3} = - 24 \sqrt{3}$

Hence, the answer is (-24√3).