Question

Which term in the expansion of ${\left\{{\left(\frac{x}{\sqrt{y}}\right)}^{1/3}+{\left(\frac{y}{x^{1/3}}\right)}^{1/2}\right\}}^{21}$ contains x and y to one and the same power?

Answer 1

Suppose T_r+1th term in the given expression contains x and y to one and the same power.
Then,

$$\begin{gathered} T_{r+1}thtermis \\ {}^{21}C_r{\left[{\left(\frac{x}{\sqrt{y}}\right)}^{1/3}\right]}^{21-r}{\left[{\left(\frac{y}{x^{1/3}}\right)}^{{}^{1/2}}\right]}^r \\ ={}^{21}C_r\left(\frac{x^{(21-r)/3}}{x^{r/6}}\right)\left(\frac{y^{r/2}}{y^{(21-r)/6}}\right) \\ ={}^{21}C_r{\left(x\right)}^{7-r/2}{\left(y\right)}^{2r/3-7/2} \\ \mathrm{Now},\mathrm{if}x\mathrm{and}y\mathrm{have}\mathrm{the}\mathrm{same}\mathrm{power},\mathrm{then} \\ 7-\frac{r}{2}=\frac{2r}{3}-\frac{7}{2} \\ \Rightarrow \frac{2r}{3}+\frac{r}{2}=7+\frac{7}{2} \\ \Rightarrow \frac{7r}{6}=\frac{21}{2} \\ \Rightarrow r=9 \\ \mathrm{Hence},\mathrm{the}\mathrm{required}\mathrm{term}\mathrm{is}\mathrm{the}10\mathrm{th}\mathrm{term} \end{gathered}$$