Question

pls solve this problem​

Answer 1

Question:

$\sf If \ x \ = \ 1 + \sqrt{2} \ \ find \ the \ value \ of \ \left( x - \dfrac{1}{x} \right)^3$

Given:

$\sf x = 1 + \sqrt{2}$

To Find:

$\sf Value \ of \ \left( x - \dfrac{1}{x} \right)^3$

Solution:

$\Longrightarrow \sf \left(x - \dfrac{1}{x} \right)^3\\ \\ \\\sf Substitute \ the \ value \ of \ x = 1 + \sqrt{2}\\ \\ \\\Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1}{1 + \sqrt{2}}\right)^3\\ \\ \\ \\ \sf Rationalizing \ the \ denominator \ we \ get,$

$\Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1}{1 + \sqrt{2}} \ \dfrac{\left(1 - \sqrt{2}\right)}{\left(1 - \sqrt{2}\right)}\right)^3\\ \\ \\ \\\Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1 - \sqrt{2}}{(1 + \sqrt{2})(1 - \sqrt{2})}\right)^3\\ \\ \\ \\\sf Using \ the \ identity \ (a + b)(a - b) = a^2 - b^2\\ \\ \\ \Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1 - \sqrt{2}}{\left(1\right)^2 - \left(\sqrt{2}\right)^2}\right)^3$

$\Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1 - \sqrt{2}}{1 - 2}\right)^3\\ \\ \\ \\\Longrightarrow \sf \left(1 + \sqrt{2} \ - \dfrac{1 - \sqrt{2}}{-1 \ }\right)^3$

$\displaystyle \Longrightarrow \sf \Bigg(1 + \sqrt{2} \ - \Big(-\left(1 - \sqrt{2} \ \right) \Big) \Bigg)^3$

$\displaystyle \Longrightarrow \sf \Bigg(1 + \sqrt{2} \ - \Big( -1 + \sqrt{2} \ \right) \Big) \Bigg)^3$

$\Longrightarrow \sf \Bigg(1 \ + \ \sqrt{2} \ + \ 1 \ - \sqrt{2} \Bigg)^3$

$\Longrightarrow \sf \Big(1 + 1 \Big)^3\\ \\ \\ \\\Longrightarrow \sf \Big(2\Big)^3\\ \\ \\ \\\Longrightarrow \sf \ \red{8}\\ \\ \\ \\\underline{\large{\boxed{\sf Answer: \left( x - \dfrac{1}{x} \right)^3 = \ \red{8} \ }}}$

Answer 2

Answer:$\huge\underline{ \underline{ \mathbb{ { \blue{ AN{ \pink{SW{ \purple{ER \: : = }}}}} }}}}$

your answer is 8...

hope you asked to only just to confirm you answer