If x be so small that its square and higher powers may be neglected, find the value of :
$\frac{\sqrt[4]{1 - \frac{3}{5} x} + {(1 + \frac{5}{6} x)}^{- 6}}{\sqrt[5]{1 + 2 x} + \sqrt[5]{1 - \frac{x}{2}}}$

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Solution

$⟹ \frac{(1 - \frac{3}{5} \times \frac{1}{4} x) + (1 + \frac{5}{6} \times x (- 6))}{(1 + \frac{2}{5} x) + (1 - \frac{x}{10})}$
$⟹ \frac{2 - \frac{103}{20} x}{2 + \frac{3}{10} x}$
$⟹ (2 - \frac{103}{20} x) \frac{1}{2} {(1 + \frac{3}{20} x)}^{- 1}$
$⟹ \frac{1}{2} (2 - \frac{103}{20} x) (1 - \frac{3}{20} x)$
$⟹ \frac{1}{2} (2 - \frac{109}{20} x) = 1 - \frac{109}{40} x$

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