Use the rule $\displaystyle{{\left({a}{b}\right)}}^{{x}}={{a}}^{{x}}\cdot{{b}}^{{x}}\usepackage{cmbright}\Huge\$ to rewrite:

$\displaystyle\frac{{{36}{{a}}^{{6}}{b}}}{{{{a}}^{{{2}}}{{144}}^{{\frac{{1}}{{2}}}}{{b}}^{{\frac{{1}}{{2}}}}}}=\frac{{{36}{{a}}^{{6}}{b}}}{{{{a}}^{{2}}{{\left({{12}}^{{2}}\right)}}^{{\frac{{1}}{{2}}}}{{b}}^{{\frac{{1}}{{2}}}}}}=\frac{{{36}{{a}}^{{6}}{b}}}{{{{12}}^{{\frac{{2}}{{2}}}}{{a}}^{{2}}{{b}}^{{\frac{{1}}{{2}}}}}}\usepackage{cmbright}\Huge\$

Rewrite as a product: $\displaystyle\frac{{36}}{{12}}\cdot\frac{{{a}}^{{6}}}{{{a}}^{{2}}}\cdot\frac{{b}}{{{b}}^{{\frac{{1}}{{2}}}}}={3}\cdot{{a}}^{{{6}-{2}}}\cdot{{b}}^{{{1}-\frac{{1}}{{2}}}}={3}{{a}}^{{4}}{{b}}^{{\frac{{1}}{{2}}}}\usepackage{cmbright}\Huge\$

Change the last factor to radical form: $\displaystyle{3}{{a}}^{{4}}\sqrt{{{b}}}\usepackage{cmbright}\Huge\$