Question

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

Answer 1

$T_n=T_{r+1}={}^n{C}_r{x}^{n-r}{y}^r$

$={}^{21}{C}_r{\left({\left(\frac{x}{\sqrt{y}}\right)}^{\frac{1}{3}}\right)}^{21-r}{\left({\left(\frac{y}{x^{\frac{1}{3}}}\right)}^{\frac{1}{2}}\right)}^r$

$={}^{21}{C}_r\left(\frac{x^{7-\frac{r}{3}}}{y^{\frac{7}{2}-\frac{r}{6}}}\right)\frac{y^{\frac{r}{2}}}{x^{\frac{r}{6}}}=\frac{x^{7-\frac{r}{3}-\frac{r}{6}}}{y^{\frac{7}{2}-\frac{r}{6}-\frac{r}{2}}}$

$\Rightarrow x\frac{45-2r-r}{6}=y\frac{21-r-3r}{6}$

Since x and y have same power

$\frac{42-3r}{6}=\frac{-(21-4r)}{6}$

=42+21=4r+3r

63=7r

r=9

Term is 10th

$(t_n=T_{r+1})$