If the ${(r+1)}^{\text{th}}$ term in the expansion of $\left(\frac{{\displaystyle a^{\frac{{\displaystyle 1}}{{\displaystyle 3}}}}}{{\displaystyle b^{\frac{{\displaystyle 1}}{{\displaystyle 6}}}}}+\frac{{\displaystyle b^{\frac{{\displaystyle 1}}{{\displaystyle 2}}}}}{{\displaystyle a^{\frac{{\displaystyle 1}}{{\displaystyle 6}}}}}\right)$ has equal exponents of both a and b, then the value of r is 

$T_{r+1}={}^{21}{\text{C}}_r{\left(\frac{{\displaystyle a^{\frac{{\displaystyle 1}}{{\displaystyle 3}}}}}{{\displaystyle b^{\frac{{\displaystyle 1}}{{\displaystyle 6}}}}}\right)}^{21-r}{\left(\frac{{\displaystyle b^{\frac{{\displaystyle 1}}{{\displaystyle 2}}}}}{{\displaystyle a^{\frac{{\displaystyle 1}}{{\displaystyle 6}}}}}\right)}^r$

$={}^{21}{\text{C}}_r\frac{a^{\frac{21-r}{3}}}{b^{\frac{21-r}{6}}}.\frac{b^{\frac{r}{2}}}{a^{\frac{r}{6}}}$

$={}^{21}{\text{C}}_{r}a\left(\frac{{\displaystyle 21-r}}{{\displaystyle 3}}\right)-\left(\frac{{\displaystyle r}}{{\displaystyle 6}}\right).b^{\left(\frac{r}{2}\right)-\left(\frac{21-r}{6}\right)}$

$={}^{21}{\text{C}}_r{a}^{\frac{42-3r}{6}}{b}^{\frac{3r-21+r}{6}}$

$={}^{21}{\text{C}}_r{a}^{\frac{42-3r}{6}}{b}^{\frac{4r-21}{6}}$

Given power of a and b are equal

$\begin{array}{l}\Rightarrow \frac{42-3r}{6}=\frac{4r-21}{6}\\ \Rightarrow 42+21=4r+3r\\ 7r=63\\ r=9\end{array}$