‘Can be solved by most 13-year olds in China’-4

In order to make this question easier to solve, we should avoid finding the actual value of $m$ as it ends up difficult to compute. Therefore we can break it down:
$m^4+\frac{1}{m^4}=\color{#D61F06}m^4+2\cdot m^4\cdot \frac{1}{m^4} +\frac{1}{m^4}\color{#333333}-2=\color{#D61F06}(m^2+\frac{1}{m^2})^2\color{#333333}-2.$ See? The steps in red used the complete square formula to break the expression into terms of lower degrees. Next we need only find $m^2+\frac{1}{m^2}$ .
$m^2+\frac{1}{m^2}=m^2-2+\frac{1}{m^2}+2=(m-\frac{1}{m})^2+2=3^2+2=11.$ Plug this into the first equation to get $m^4+\frac{1}{m^4}=(m^2+\frac{1}{m^2})^2-2=11^2-2=119$ .