You may have correctly rewritten the expression as:

$\displaystyle\frac{{{{s}}^{{3}}{{\left({36}{{r}}^{{9}}{{s}}^{{4}}\right)}}^{{\frac{{1}}{{2}}}}}}{{{4}{s}{{\left({27}{{r}}^{{12}}\right)}}^{{\frac{{1}}{{3}}}}}}=\frac{{{{s}}^{{3}}\cdot{{\left({{6}}^{{2}}\right)}}^{{\frac{{1}}{{2}}}}\cdot{{r}}^{{\frac{{9}}{{2}}}}\cdot{{s}}^{{\frac{{4}}{{2}}}}}}{{{4}\cdot{s}\cdot{{\left({{3}}^{{3}}\right)}}^{{\frac{{1}}{{3}}}}\cdot{{r}}^{{\frac{{12}}{{3}}}}}}\usepackage{cmbright}\Huge\$

and simplified this down to $\displaystyle\frac{{6}}{{12}}\cdot\frac{{{r}}^{{\frac{{9}}{{2}}}}}{{{r}}^{{4}}}\cdot\frac{{{s}}^{{5}}}{{s}}\usepackage{cmbright}\Huge\$ , but then you divided the exponents instead of subtracting them.