Powerful Exponents

$\begin{cases} 60^a = 3 & \implies a \log 60 = \log 3 & \implies a = \dfrac {\log 3}{\log 60} \\ 60^b = 5 & \implies b \log 60 = \log 5 & \implies b = \dfrac {\log 5}{\log 60} \end{cases}$

$\begin{aligned} x & = 12^{\frac {1-a-b}{2(1-b)}} \\ \log x & = \frac {1-a-b}{2(1-b)} \log 12 \\ \frac {\log x}{\log 12} & = \frac {1-\frac {\log 3}{\log 60} - \frac {\log 5}{\log 60}}{2\left(1-\frac {\log 5}{\log 60}\right)} \\ & = \frac {\log 60 - \log 3 - \log 5}{2(\log 60 - \log 5)} \\ & = \frac {\log \frac {60}{3 \times 5}}{2 \times \log \frac {60}5} \\ & = \frac {\log 4}{2 \times \log 12} \\ \implies \log x & = \frac {\log 4 \times \log 12}{2 \times \log 12} = \frac {2 \log 2}2 = \log 2 \\ \implies x & = \boxed{2} \end{aligned}$