A physical quantity $P$ is related to four observables $a$, $b$, $c$ and $d$ as $P = \frac{\sqrt{ab} \cdot d^{\alpha}}{\sqrt{c}}$ ($\alpha$ is constant). The percentage errors in $a$, $b$, $c$ and $d$ are $0.5\%$ in each. If the percentage error in $P$ is $2\%$, then $\alpha$ is