If $x$ be so small that its square and higher powers may be neglected, find the value of :
$(1 - 7 x)^{\frac{1}{3}} (1 + 2 x)^{- \frac{3}{4}}$

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Solution

let
$y = (1 - 7 x)^{\frac{1}{3}} (1 + 2 x)^{\frac{- 3}{4}}$

$= (1 - \frac{1}{3} .7 x - \frac{2}{9} \frac{49 x^{2}}{2!} . . . . .) (1 - \frac{3}{4} .2 x + \frac{21}{16} \frac{4 x^{2}}{2!} . . .)$

x be so small that its square and higher powers may be neglected

$= (1 - \frac{1}{3} .7 x) (1 - \frac{3}{2} . x)$

again neglect square terms

$= 1 - \frac{23 x}{6}$

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