Question

$<marquee>please solve this$​

Answer 1

$\large{\mathfrak{\overbrace{\underbrace{\pink{\fbox{\green{\blue{\bf\:S}\pink{o}\red{l}\orange{u}\purple{t}\blue{i}\green{o}\red{n}}}}}}}}$

$\red\longmapsto\dfrac{2}{\sqrt[3]{9}-\sqrt[3]{3} + 1} - \dfrac{1}{\sqrt[3]{9} + \sqrt[3]{3} + 1}$

$\red\longmapsto\dfrac{2}{(\sqrt[3]{9} + 1)-\sqrt[3]{3}} - \dfrac{1}{(\sqrt[3]{9} +1) + \sqrt[3]{3}}$

$\blue{\textbf{Taking LCM of Denominator and using (a + b)(a - b) = (a² - b²) Now,}}$

$\red\longmapsto\dfrac{2( \sqrt[3]{9} + \sqrt[3]{3} + 1) - 1( \sqrt[3]{9} - \sqrt[3]{3} + 1) }{( \sqrt[3]{9} + 1)^{2} - (\sqrt[3]{3})^{2} }$

$\green{\bf \: using \: \sqrt[a]{b} \: as \: {b}^{ \frac{1}{a}} \: Now,}$

$\red\longmapsto\dfrac{2 \times {9}^{\frac{1}{3}} + 2 \times {3}^{ \frac{1}{3}} + 2 - {9}^{ \frac{1}{3}} + {3}^{ \frac{1}{3}} - 1}{(1 + {9}^{ \frac{1}{3}}) ^{2} -(3^{ \frac{1}{3}})^{2}}$

$\pink{\textsf{Now using (a + b)² = a² + b² + 2ab in Denominator ,}}$

$\red\longmapsto\dfrac{9^{\frac{1}{3}} + 3 \times {3}^{ \frac{1}{3}} + 1}{1 + {9}^{ \frac{2}{3}} + 2 \times {9}^{ \frac{1}{3}} - {3}^{ \frac{2}{3}}}$

$\red\longmapsto \: \dfrac{ {9}^{ \frac{1}{3}} + {3}^{ \frac{4}{3}} + 1}{1 + {9}^{ \frac{2}{3}} + {3}^{\frac{2}{3}}}$

$\red\longmapsto \: \dfrac{ {3}^{ \frac{2}{3} } + {3}^{ \frac{4}{3} } + 1 }{{3}^{ \frac{2}{3} } + {3}^{ \frac{4}{3} } + 1}$

$\red\longmapsto\:\rm \: \bold{\boxed{\large{\boxed{\orange{\small{\boxed{\large{\red{\bold{1}}}}}}}}}}$

$\rule{200}{4}$