Question

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

Answer 1

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

since it is given that

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

so we expand each bracket upto single power only

$=[1+\frac{2x}{3}(-5\left)\right]\times 2(1+\frac{3x}{4}\times \frac{1}{2})\times \frac{1}{8}[1+\frac{x}{4}(-\frac{3}{2}\left)\right]$

again neglecting square terms

$=\frac{1}{4}[1-\frac{10x}{3}]\left[1\right]$

$=\frac{1}{4}-\frac{5x}{6}$