If $|x| \lt \frac{2}{2}$ then the $4^{ ext {th }}$ term in the expansion of $(3 x-2)^{2 / 3}$ is

Question

If $|x| \lt \frac{2}{2}$ then the $4^{	ext {th }}$ term in the expansion of $(3 x-2)^{2 / 3}$ is

Aicharya · Accepted Answer

$(3 x-2)^{\frac{2}{3}}=\left[-2\left(1-\frac{3 x}{2}ight)ight]^{\frac{2}{3}}=(-2)^{\frac{2}{3}}\left(1-\frac{3 x}{2}ight)^{\frac{2}{3}}$ $\begin{aligned} & =\sqrt[3]{4}\left(1-\frac{3 x}{2}ight)^{\frac{2}{3}} \ & T_4=-\sqrt[3]{4} \frac{\frac{2}{3}\left(\frac{2}{3}-1ight)\left(\frac{2}{3}-2ight)}{3!}\left(\frac{3 x}{2}ight)^3 \ & =-\sqrt[3]{4} 	imes \frac{1}{6} 	imes \frac{2}{3} 	imes\left(\frac{-1}{3}ight)\left(\frac{-4}{3}ight) 	imes \frac{27}{8} x^3=-\frac{\sqrt[3]{4}}{6} x^3\end{aligned}$

If $|x| \lt \frac{2}{2}$ then the $4^{\text {th }}$ term in the expansion of $(3 x-2)^{2 / 3}$ is

Solution