Question

If x equal to 2 - root 3, find the value of ( x - 1 by x) whole cube

Answer 1

$\underline{\mathfrak{\huge{Question:}}}$

Given is that : x = $\tt{2 - \sqrt{3}}$

Now, find the value of : $\tt{{x - \frac{1}{x}}^{3}}$

$\underline{\mathfrak{\huge{Answer:}}}$

Start with simplifying the "$\tt{x - \frac{1}{x}}$" term :-

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

Start rationalizing it now :-

=》 $2 - \sqrt{3} - (\frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}})$

Some more steps to go :-

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

Few more steps :-

=》 $2 - \sqrt{3} - (\frac{2 + \sqrt{3}}{4-3}$

Almost done with the result :-

=》 $2 - \sqrt{3} - (2 + \sqrt{3})$

Open the brackets now :-

=》 $2 - \sqrt{3} - 2 - \sqrt{3}$

Do the necessary operations and Get your final result :-

=》 $(-2\sqrt{3})$

Now, we can easily put the value of $x - \frac{1}{x}$ in the equation :-

${x-\frac{1}{x}}^{3}$

Putting it there, we get :-

=》 $(-2\sqrt{3})^{3}$

The final answer therefore, will be :-

=》 $\tt{(-24\sqrt{3})}$

There's your answer !