Question

if both a and b are rational number.. find a and b

$\tt\frac{ \sqrt{2} + \sqrt{3} }{ \sqrt[3]{2} - \sqrt[2]{3} } = a - \sqrt[b]{6}$
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Answer 1

AnsweR :

$\pink { \boxed{\sf \large \purple{ \: a = 2 \: and \: b = - 5/6}}}$

SolutioN :

Multiplying the numerator and denominator by rationalisation factor of the denominator.

$\sf = \frac{ \sqrt{2} + \sqrt[2]{3} }{ \sqrt[3]{2} - \sqrt[2]{3} } \times \frac{ \sqrt[3]{2} + \sqrt[2]{3} }{ \sqrt[3]{2} + \sqrt[2]{3} } \\ \\ \sf = \frac{ (\sqrt{2} + \sqrt{3} )( \sqrt[3]{2} + \sqrt[2]{3} ) }{ (\sqrt[3]{2} - \sqrt[2]{3} )( \sqrt[3]{2} + \sqrt[2]{3}) } \\ \\ \sf = \frac{ \sqrt{2} \times \sqrt[3]{2} + \sqrt{2} \times \sqrt[2]{3} + \sqrt{3} \times \sqrt[3]{2} + \sqrt{3} \times \sqrt[2]{3} }{ {( \sqrt[3]{2}) }^{2} - {( \sqrt[2]{3}) }^{2} } \\ \\ \sf = \frac{ \sqrt[3]{2 \times 2} + \sqrt[2]{3 \times 2} + \sqrt[3]{3 \times 2} + \sqrt[2]{3 \times 3} }{9 {( \sqrt{2)} - 4 {( \sqrt{3} )}^{2} } } \\ \\ \sf = \frac{3 \times 2 + \sqrt[2]{6} + \sqrt[3]{6} + 2 \times 3 }{9 \times 2 - 4 \times 3} \\ \\ \sf = \frac{6 + (2 + 3) \sqrt{6} + 6 }{18 - 12} = \frac{12 + \sqrt[5]{6} }{6} = 2 + \frac{5}{6} \sqrt{6}$

$\sf \therefore \frac{ \sqrt{2} + \sqrt{3} }{ \sqrt[3]{2} - \sqrt[2]{3} } = a - b \sqrt{6}$

$\sf = 2 + \frac{5}{6} \sqrt{6} = a - b \sqrt{6} \\ \\ \sf = a - b \sqrt{6} = 2 - ( - 5/6) \sqrt{6} \\ \\ \sf \large \bold{a = 2 \: and \: b = - 5/6}$

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Answer 2

AnsweR :

$\pink { \boxed{\sf \large \purple{ \: a = 2 \: and \: b = - 5/6}}}$

SolutioN :

Multiplying the numerator and denominator by rationalisation factor of the denominator.

$\sf = \frac{ \sqrt{2} + \sqrt[2]{3} }{ \sqrt[3]{2} - \sqrt[2]{3} } \times \frac{ \sqrt[3]{2} + \sqrt[2]{3} }{ \sqrt[3]{2} + \sqrt[2]{3} } \\ \\ \sf = \frac{ (\sqrt{2} + \sqrt{3} )( \sqrt[3]{2} + \sqrt[2]{3} ) }{ (\sqrt[3]{2} - \sqrt[2]{3} )( \sqrt[3]{2} + \sqrt[2]{3}) } \\ \\ \sf = \frac{ \sqrt{2} \times \sqrt[3]{2} + \sqrt{2} \times \sqrt[2]{3} + \sqrt{3} \times \sqrt[3]{2} + \sqrt{3} \times \sqrt[2]{3} }{ {( \sqrt[3]{2}) }^{2} - {( \sqrt[2]{3}) }^{2} } \\ \\ \sf = \frac{ \sqrt[3]{2 \times 2} + \sqrt[2]{3 \times 2} + \sqrt[3]{3 \times 2} + \sqrt[2]{3 \times 3} }{9 {( \sqrt{2)} - 4 {( \sqrt{3} )}^{2} } } \\ \\ \sf = \frac{3 \times 2 + \sqrt[2]{6} + \sqrt[3]{6} + 2 \times 3 }{9 \times 2 - 4 \times 3} \\ \\ \sf = \frac{6 + (2 + 3) \sqrt{6} + 6 }{18 - 12} = \frac{12 + \sqrt[5]{6} }{6} = 2 + \frac{5}{6} \sqrt{6}$

$\sf \therefore \frac{ \sqrt{2} + \sqrt{3} }{ \sqrt[3]{2} - \sqrt[2]{3} } = a - b \sqrt{6}$

$\sf = 2 + \frac{5}{6} \sqrt{6} = a - b \sqrt{6} \\ \\ \sf = a - b \sqrt{6} = 2 - ( - 5/6) \sqrt{6} \\ \\ \sf \large \bold{a = 2 \: and \: b = - 5/6}$

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