Question

Find the absolute and percent relative uncertainty and express each answer with a reasonable number of significant figures.

$$\begin{gathered} \mathrm{a})9.23\left(\pm 0.03\right)+4.21\left(\pm 0.02\right)-3.26\left(\pm 0.06\right)=? \\ \mathrm{b})91.3\left(\pm 1.0\right)\times 40.3\left(\pm 0.02\right)/21.1\left(\pm 0.02\right)=? \\ \mathrm{c})\left[4.97\left(\pm 0.05\right)-1.86\left(\pm 0.01\right)\right]/21.1\left(\pm 0.02\right)=? \\ \mathrm{d})2.0164\left(\pm 0.0008\right)+1.233\left(\pm 0.002\right)+4.61\left(\pm 0.01\right)=? \\ \mathrm{e})2.0164\left(\pm 0.0008\right)\times {10}^3+1.233\left(\pm 0.002\right)\times {10}^2+4.61\left(\pm 0.01\right)\times {10}^1=? \\ \mathrm{f}){\left[3.14\left(\pm 0.05\right)\right]}^{1/3}=? \\ \mathrm{g})\log {\left[3.14\left(\pm 0.05\right)\right]}^{1/3}=? \end{gathered}$$

Short answer

To calculate the absolute and percent relative uncertainty

Step-by-step solution

To calculate the absolute and percent relative uncertainty.

$$\begin{gathered} \mathrm{a})9.23\left(\pm 0.03\right)+4.21\left(\pm 0.02\right)-3.26\left(\pm 0.06\right)=10.18\left(\pm 0.07\mathrm{or}\pm \mathbf{0}\mathbf{.}\mathbf{07}\mathbf{\%}\right) \\ \mathrm{b})91.3\left(\pm 1.0\right)\times 40.3\left(\pm 0.02\right)/21.1\left(\pm 0.02\right)=174\left(\pm 3\mathrm{or}\pm 2\%\right)\mathrm{or}174.4\left(\pm 2.7\mathrm{or}\mathbf{\pm }\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\%}\right) \\ \mathrm{c})\left[4.97\left(\pm 0.05\right)-1.86\left(\pm 0.01\right)\right]/21.1\left(\pm 0.02\right)=0.147\left(\pm 0.0003\mathrm{or}\mathbf{\pm }\mathbf{2}\mathbf{\%}\right)\mathrm{or}0.1474\left(\pm 0.0028\right)\left(\pm \mathbf{1}\mathbf{.}\mathbf{9}\mathbf{\%}\right) \\ \mathrm{d})2.0164\left(\pm 0.0008\right)+1.233\left(\pm 0.002\right)+4.61\left(\pm 0.01\right)=7.86\left(\pm 0.01\mathrm{or}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{\%}\right) \\ \mathrm{e})2.0164\left(\pm 0.0008\right)\times {10}^3+1.233\left(\pm 0.002\right)\times {10}^2+4.61\left(\pm 0.01\right)\times {10}^1=2185.8\left(\pm 0.8\mathrm{or}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{\%}\right) \\ \mathrm{f})\left[3.14\left(\pm 0.05\right)\right]{\mathrm{a}}^{1/3}=1.4643\left(\pm 0.0078\mathrm{or}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{53}\mathbf{\%}\right) \\ \mathrm{g})\log {\left[3.14\left(\pm 0.05\right)\right]}^{1/3}=0.49469\left(0.0069\mathrm{or}\mathbf{1}\mathbf{.}\mathbf{39}\mathbf{\%}\right) \end{gathered}$$

Formula to calculate the percent relative uncertainty 

The percent relative uncertainty is calculated with the formula (e represents the $\pm$value)

$\mathbf{=}\sqrt{{\mathbf{e}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{e}}_{\mathbf{2}}^{\mathbf{2}}\mathbf{+}{\mathbf{e}}_{\mathbf{3}}^{\mathbf{2}}}$