Question

If x be so small that its square and higher powers may be neglected, find the value of :
$\frac{\sqrt[3]{38+3x}-\sqrt[5]{51-x}}{(1+5x{)}^{\frac{3}{5}}+{(4+\frac{x}{2})}^{\frac{1}{2}}}$

Answer 1

$⟹\frac{2(1+\frac{3x}{8}\times \frac{1}{3})-(1-x\times \frac{5}{6})}{(1+5x\times \frac{3}{5})+2(1+\frac{1}{4}\times \frac{x}{2}\times \frac{1}{2})}$
$⟹\frac{2+\frac{x}{4}-1+\frac{x}{5}}{1+3x+2+\frac{x}{8}}$
$⟹(1+\frac{9}{20}x){(3+\frac{25}{8}x)}^{-1}$
$⟹(1+\frac{9}{20}x)\left(\frac{1}{3}\right){(1+\frac{25}{24}x)}^{-1}$
$⟹\frac{1}{3}(1+\frac{9x}{20})(1-\frac{25x}{24})$
$⟹\frac{1}{3}(1-\frac{71}{120}x)$
$=\frac{1}{3}-\frac{71}{360}x$