The quantity $ Q $ is expressed as $ Q = X^{-2}Y^{+\frac{3}{2}}Z^{-\frac{2}{5}} $. The variables $ X $, $ Y $, and $ Z $ are independent parameters with fractional errors of 0.1, 0.2, and 0.5, respectively. To determine the maximum fractional error in $ Q $, we can use the following method:

$ \text{Fractional error in } Q = \left| -2 \right| \frac{\Delta X}{X} + \left| \frac{3}{2} \right| \frac{\Delta Y}{Y} + \left| -\frac{2}{5} \right| \frac{\Delta Z}{Z} $

Substituting the fractional errors into the equation:

$ = 2 \times 0.1 + \frac{3}{2} \times 0.2 + \frac{2}{5} \times 0.5 $

Calculating each term gives:

$ = 0.2 + 0.3 + 0.2 $

Summing these values results in:

$ = 0.7 $

Thus, the maximum fractional error in $ Q $ is 0.7.