Summing the alphabets !

Inside the radical, we first simplify the exponent of $\displaystyle \frac{1}{\sqrt{27}}$ . We have $\displaystyle 2 - \frac{\log_{5}{13}}{2\log_{5}{9}} \Longrightarrow 2 - \frac{\log_{5}{13}}{\log_{5}{81}} \Longrightarrow 2 - \log_{81}{13} \Longrightarrow \color{#D61F06}{2 - \frac{1}{4}\log_{3}{13}}.$ Then, $\displaystyle \sqrt{\left(\frac{1}{3^{\frac{3}{2}}}\right)^{\color{#D61F06}{2 - \frac{1}{4}\log_{3}{13}}}} \Longrightarrow \sqrt{\left(3^{-\frac{3}{2}}\right)^{\color{#D61F06}{2 - \frac{1}{4}\log_{3}{13}}}} \Longrightarrow \sqrt{3^{\frac{3}{8}\log_{3}{13} - 3}} \Longrightarrow \sqrt{3^{\frac{3}{8}\log_{3}{13} - \log_{3}{27}}}$ $\displaystyle \Longrightarrow \sqrt{3^{\log_{3}{\frac{13^{\frac{3}{8}}}{3^3}}}} \Longrightarrow \sqrt{\frac{13^{\frac{3}{8}}}{3^3}} = 13^{\frac{3}{16}} \times 3^{\frac{-3}{2}}.$ Thus, we get $\displaystyle a + b + c + d = -3 + 2 + 3 + 16 = 18$ .