If x = [3] ( 2 + 1) - [3] ( 2 - 1) , then x^3 + 3x = - Vidyadip

Question

If $x = \sqrt[3]{{(\sqrt 2 + 1)}} - \sqrt[3]{{(\sqrt 2 - 1)}}$, then ${x^3} + 3x = $

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Answer

a
(a) $x = {(\sqrt 2 + 1)^{1/3}} - {(\sqrt 2 - 1)^{1/3}}$

${x^3} = (\sqrt 2 + 1) - (\sqrt 2 - 1) - 3{(\sqrt 2 + 1)^{1/3}}\,{(\sqrt 2 - 1)^{1/3}}$

$\,\left[ {\sqrt[3]{{(\sqrt 2 + 1)}}\, - \sqrt[3]{{\sqrt 2 - 1}}} \right]$

${x^3} = 2 - 3\,{(2 - 1)^{1/3\,}}x$ $ \Rightarrow {x^3} + 3x = 2$.

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