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

Two resistors of resistances $R_{1} = 100 \pm 3 \ \Omega$ and $R_{2} = 200 \pm 4 \ \Omega$ are connected $(a)$ in series,$(b)$ in parallel. Find the equivalent resistance of the $(a)$ series combination,$(b)$ parallel combination. Use for $(a)$ the relation $R = R_{1} + R_{2}$ and for $(b)$ $\frac{1}{R^{\prime}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$ and $\frac{\Delta R^{\prime}}{R^{\prime 2}} = \frac{\Delta R_{1}}{R_{1}^{2}} + \frac{\Delta R_{2}}{R_{2}^{2}}$.

$A$ metal wire has mass $(0.4 \pm 0.002) \, g$,radius $(0.3 \pm 0.001) \, mm$ and length $(5 \pm 0.02) \, cm$. The maximum possible percentage error in the measurement of density will nearly be $.......\%$

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is $\left(\frac{x}{100}\right) \%$. If the relative errors in measuring the mass and the diameter are $6.0 \%$ and $1.5 \%$ respectively,the value of $x$ is

The density of a cube is measured by measuring its mass and the length of its sides. If the maximum error in the measurement of mass and length are $4\%$ and $3\%$ respectively,the maximum error in the measurement of density will be ........ $\%$

Why should the measurement of physical quantities with large exponents in an experiment be taken with high precision?