Question

please solve step by step

Answer 1

Hey

Nucleofies :p

$\sqrt[3]{ \sqrt[3]{2} - 1 }$

$= \sqrt[3]{( \sqrt[3]{2} - 1)( \frac{ \sqrt[3]{4} + \sqrt[3]{2} + 1 }{\sqrt[3]{4} + \sqrt[3]{2} + 1} ) }$

$= \sqrt[3]{ \frac{1}{\sqrt[3]{4} + \sqrt[3]{2} + 1} }$

$= \sqrt[3]{( \frac{1}{\sqrt[3]{4} + \sqrt[3]{2} + 1})( \frac{ \sqrt[3]{4} - \sqrt[3]{2} + 1 }{\sqrt[3]{4} - \sqrt[3]{2} + 1} )}$

$= \sqrt[3]{ \frac{\sqrt[3]{4} - \sqrt[3]{2} + 1}{ {( \sqrt[3]{2} + 1)}^{2} } }$

$= \frac{\sqrt[3]{4} - \sqrt[3]{2} + 1}{ \sqrt[3]{ {( \sqrt[3]{2} + 1 )^{2} {(\sqrt[3]{4} - \sqrt[3]{2} + 1)}^{2} } } }$

$= \frac{ \sqrt[3]{4} - \sqrt[3]{2} + 1}{ \sqrt[3]{9} }$

Now, put this (not) nested value of Nested Expression above :
$\sqrt[3]{ \sqrt[3]{2} - 1}$

To get :
$\sqrt[3]{a} + \sqrt[3]{b} = \sqrt[3]{ \frac{4}{9} } - \sqrt[3]{ \frac{2}{9} }$

Without loss of Generality, you can assume one value of a, b as :
$a = \frac{4}{9} \: \: | || | \: \: b = \frac{ - 2}{9}$

And hence,
$a + 2b = 0$