Question

When x is so small that its square and higher powers maybe neglected, find the value of $\frac{{(1+\frac{2}{3}x)}^{-5}+\sqrt{4+2x}}{\sqrt{(4+x{)}^3}}$

Answer 1

Since $x$ is small so $x^2$ and higher powers can be neglected, it will be sufficient to retain the first two terms in the expansion of each binomial. Therefore the expression

$=\frac{{(1+\frac{2}{3}x)}^{-5}+2{(1+\frac{x}{2})}^{\frac{1}{2}}}{8{(1+\frac{x}{4})}^{\frac{3}{2}}}$

$=\frac{(1-\frac{10}{3}x)+2(1+\frac{x}{4})}{8(1+\frac{3}{8}x)}$

$=\frac{1}{8}(3-\frac{17}{6}x){(1+\frac{3}{8}x)}^{-1}$

$=\frac{1}{8}(3-\frac{17}{6}x)(1-\frac{3}{8}x)$

$=\frac{1}{8}(3-\frac{95}{24}x)$

the terms involving $x^2$ is neglected.