A physical quantity P is related to four observables a, b, c and d...

As, $P=\frac{a^{1 / 2} \cdot b^{1 / 2} \cdot d^\alpha}{c^{1 / 2}}$ So $\%$ error in $P=\frac{\Delta P}{P} \times 100$$\begin{aligned} =\frac{1}{2}& \left(\frac{\Delta a}{a}\right) \times₁₀₀⁺\frac{1}{2}\left(\frac{\Delta b}{b} \times100\right) \\ \text{ and } +\alpha\left(\frac{\Delta d}{d} \times100\right)+\frac{1}{2}\left(\frac{\Delta c}{c} \times100\right) \end{aligned}$\Rightarrow \%$error in$P=\frac{1}{2} \times \%$error in$a+\frac{1}{2} \times \%$error in$b+\frac{1}{2} \times \%$error in$c+\alpha \times \%$error in$d$Now given, \% Error in$P=2 \%\%$Error in$a=\%$error in$b=\%$Error in$c=\%$error in$d=0.5 \%$Substituting these values in Eq. (i), we get$$\begin{aligned}2 \text{ and } =\frac{1}{2} \times0.5+\frac{1}{2} \times0.5+\frac{1}{2} \times0.5+\alpha \times0.5 \\ \Rightarrow 2 \text{ and } =\frac{1}{2}\left(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\alpha\right) \Rightarrow4=\frac{3}{2}+\alpha \end{aligned}$$or$\alpha=4-\frac{3}{2}=\frac{5}{2}\therefore \alpha$is$5 / 2$.$