Question

Expand to 4 terms the following expressions :
$(9-6x{)}^{-\frac{2}{3}}$

Answer 1

Let $y=(9-6x{)}^{\frac{-2}{3}}=9^{\frac{-2}{3}}{(1-\frac{2x}{3})}^{\frac{-2}{3}}$

Let us expand it upto $4$ terms

${(1-\frac{2x}{3})}^{\frac{-2}{3}}=1+\left(\frac{-2}{3}\right)\left(\frac{-2x}{3}\right)+\left(\frac{-2}{3}\right)(\frac{-2}{3}-1)\left(\frac{4x^2}{9.2!}\right)+\left(\frac{-2}{3}\right)(\frac{-2}{3}-1)(\frac{-2}{3}-2)\left(\frac{-8x^3}{27.3!}\right).........$

$=1+\frac{4x}{9}+\frac{20x^2}{81}+\frac{640x^3}{\mathrm{27.27.3}}...$

$\therefore$ $(9-6x{)}^{-\frac{2}{3}}=9^{-\frac{2}{3}}[1+\frac{4x}{9}+\frac{20x^2}{81}+\frac{640x^3}{\mathrm{27.27.3}}...]$