This part is far more straight forward than the last. Basically if you have a really small number like 0.000000000001, instead of writing it out this tediously we can just count the zeros and write $1 \times 10^{\fillin[11]}$. Similarly, if you have a really large number $900000000000000000$. You could just write $9\times 10^17$. It is more concise and less prone to errors. \begin{boxdef} \textbf{Tedious}: \begin{solutionordottedlines}[1cm] annoying to do; repetitive \end{solutionordottedlines} \end{boxdef} \begin{boxdef} \textbf{Concise}: \begin{solutionordottedlines}[1cm] short. less lengthy but equally effective \end{solutionordottedlines} \end{boxdef} \begin{boxdef} \textbf{Prone}: \begin{solutionordottedlines}[1cm] sensitive to. \end{solutionordottedlines} \end{boxdef} \begin{examplebox} \subsection{Examples:} \begin{questions} \Question[8] Write in scientific notation. \begin{multicols}{2} \begin{parts} \part 610 = \fillin[$6.1\times 10^2$] \part 21000 = \fillin[$2.1\times 10^4$] \part 0.0067 = \fillin[$4.6\times 10^7$] \part 0.00002 = \fillin[$8.1\times 10^1$] \part 46000000 = \fillin[$6.7\times 10^{-3}$] \part 81 = \fillin[$2\times 10^{-5}$] \part 0.07 = \fillin[$7\times 10^{-2}$] \part 8.17 = \fillin[$8.17\times 10^0$] \end{parts} \end{multicols} \Question[4] Now go the other way; write the following in decimal form: \begin{multicols}{2} \begin{parts} \part \(2.1 \times 10^{3}\) = \fillin[2100] \part \(6.3 \times 10^{5}\) = \fillin[630000] \part \(5 \times 10^{-4}\) = \fillin[0.0005] \part \(8.12 \times 10^{-2}\) = \fillin[0.0812] \end{parts} \end{multicols} \Question[4] Simplify and write in scientific notation. \begin{parts} \part \(\left(3 \times 10^{4}\right) \times\left(2 \times 10^{6}\right)\) \begin{solutionordottedlines} \(\left(3 \times 10^{4}\right) \times\left(2 \times 10^{6}\right)=3 \times 10^{4} \times 2 \times 10^{6}\) \[ \begin{aligned} & =3 \times 2 \times 10^{4} \times 10^{6} \\ & =6 \times 10^{10} \end{aligned} \] \end{solutionordottedlines} \part \(\left(9 \times 10^{7}\right) \div\left(3 \times 10^{4}\right)\) \begin{solutionordottedlines} \(\left(9 \times 10^{7}\right) \div\left(3 \times 10^{4}\right)=\frac{9 \times 10^{7}}{3 \times 10^{4}}\) \[ \begin{aligned} & =\frac{9}{3} \times \frac{10^{7}}{10^{4}} \\ & =3 \times 10^{3} \end{aligned} \] \end{solutionordottedlines} \part \(\left(4.1 \times 10^{4}\right)^{2}\) \begin{solutionordottedlines} \(\left(4.1 \times 10^{4}\right)^{2}=4.1^{2} \times\left(10^{4}\right)^{2}\) \[ \begin{aligned} & =16.81 \times 10^{8} \\ & =1.681 \times 10^{9} \end{aligned} \] \end{solutionordottedlines} \part \(\left(2 \times 10^{5}\right)^{-2}\) \begin{solutionordottedlines} \(\left(2 \times 10^{5}\right)^{-2}=2^{-2} \times\left(10^{5}\right)^{-2}\) \[ \begin{aligned} & =\frac{1}{2^{2}} \times 10^{-10} \\ & =0.25 \times 10^{-10} \\ & =2.5 \times 10^{-11} \end{aligned} \] \end{solutionordottedlines} \end{parts} \end{questions} \end{examplebox} \begin{exercisebox} \subsection{Exercises:} \begin{questions} \Question[6] Write as a power of 10. \begin{multicols}{2}\begin{parts} \part \(\frac{1}{10} = \fillin[$10^{-1}$]\) \part \(\frac{1}{100} = \fillin[$10^{-2}$]\) \part \(\frac{1}{1000} = \fillin[$10^{-3}$]\) \part 1 trillionth = \fillin[$10^{-12}$] \part \(\frac{1}{100000} = \fillin[$10^{-5}$]\) \part 1 millionth = \fillin[$10^{-6}$] \end{parts}\end{multicols} \Question[8] Write in scientific notation. \begin{multicols}{2}\begin{parts} \part 510 = \fillin[$5.1 \times 10^2$] \part 5300 = \fillin[$5.3 \times 10^3$] \part 796000000 = \fillin[$7.96 \times 10^8$] \part 576000000000 = \fillin[$5.76 \times 10^{11}$] \part 0.008 = \fillin[$8 \times 10^{-3}$] \part 0.06 = \fillin[$6 \times 10^{-2}$] \part 0.000041 = \fillin[$4.1 \times 10^{-5}$] \part 0.000000006 = \fillin[$6 \times 10^{-9}$] \end{parts}\end{multicols} \Question[8] Write in decimal form: \begin{multicols}{2}\begin{parts} \part \(3.24 \times 10^{4} = \fillin[32400]\) \part \(7.2 \times 10^{3} = \fillin[7200]\) \part \(2.7 \times 10^{6} = \fillin[2700000]\) \part \(5.1 \times 10^{0} = \fillin[5.1]\) \part \(5.6 \times 10^{-2} = \fillin[0.056]\) \part \(1.7 \times 10^{-3} = \fillin[0.0017]\) \part \(2.01 \times 10^{-3} = \fillin[0.00201]\) \part \(9.7 \times 10^{-1} = \fillin[0.97]\) \end{parts}\end{multicols} \Question[1] Light travels approximately \(299000 \mathrm{~km}\) in a second. Express this in scientif notation. \begin{solutionordottedlines}[2cm] \(2.99 \times 10^5 \mathrm{~km/s}\) \end{solutionordottedlines} \Question[2] The mass of a copper sample is \(0.0089 \mathrm{~kg}\). Express this in scientific notation. \begin{solutionordottedlines}[2cm] \(8.9 \times 10^{-3} \mathrm{~kg}\) \end{solutionordottedlines} \Question[2] The distance between interconnecting lines on a silicon chip for a computer is approximately \(0.00000004 \mathrm{~m}\). Express this in scientific notation. \begin{solutionordottedlines}[2cm] \(4 \times 10^{-8} \mathrm{~m}\) \end{solutionordottedlines} \Question[8] Simplify, expressing the answer in scientific notation. \begin{multicols}{2}\begin{parts} \part \(\left(4 \times 10^{5}\right) \times\left(2 \times 10^{6}\right)\) \begin{solutionordottedlines}[2cm] \(8 \times 10^{11}\) \end{solutionordottedlines} \part \(\left(2.1 \times 10^{6}\right) \times\left(3 \times 10^{7}\right)\) \begin{solutionordottedlines}[2cm] \(6.3 \times 10^{13}\) \end{solutionordottedlines} \part \(\left(4 \times 10^{2}\right) \times\left(5 \times 10^{-7}\right)\) \begin{solutionordottedlines}[2cm] \(2 \times 10^{-4}\) \end{solutionordottedlines} \part \(\left(3 \times 10^{6}\right) \times\left(8 \times 10^{-3}\right)\) \begin{solutionordottedlines}[2cm] \(2.4 \times 10^{4}\) \end{solutionordottedlines} \part \(\left(5 \times 10^{4}\right) \div\left(2 \times 10^{3}\right)\) \begin{solutionordottedlines}[2cm] \(2.5 \times 10^{1}\) \end{solutionordottedlines} \part \(\left(8 \times 10^{9}\right) \div\left(4 \times 10^{3}\right)\) \begin{solutionordottedlines}[2cm] \(2 \times 10^{6}\) \end{solutionordottedlines} \part \(\left(6 \times 10^{-4}\right) \div\left(8 \times 10^{-5}\right)\) \begin{solutionordottedlines}[2cm] \(7.5 \times 10^{0}\) \end{solutionordottedlines} \part \(\left(1.2 \times 10^{6}\right) \div\left(4 \times 10^{7}\right)\) \begin{solutionordottedlines}[2cm] \(3 \times 10^{-2}\) \end{solutionordottedlines} \end{parts}\end{multicols} \Question[3] If the average distance from the Earth to the Sun is \(1.4951 \times 10^{8} \mathrm{~km}\) and light travels at \(3 \times 10^{5} \mathrm{~km} / \mathrm{s}\), how long does it take light to travel from the Sun to the Earth? \begin{solutionordottedlines}[2cm] \[ \begin{aligned} & \text{Time} = \frac{\text{Distance}}{\text{Speed}} \\ & = \frac{1.4951 \times 10^{8} \mathrm{~km}}{3 \times 10^{5} \mathrm{~km/s}} \\ & = 4.98367 \times 10^{2} \mathrm{~s} \\ & = 4.98367 \times 10^{2} \mathrm{~s} \approx 498.367 \mathrm{~s} \end{aligned} \] \end{solutionordottedlines} \Question[3] The furthest galaxy detected by optical telescopes is approximately \(4.6 \times 10^{9}\) light years from us. How far is this in kilometres? (Light travels at \(3 \times 10^{5} \mathrm{~km} / \mathrm{s}\).) \begin{solutionordottedlines}[2cm] \[ \begin{aligned} & \text{Distance in km} = 4.6 \times 10^{9} \text{ light years} \times \frac{9.461 \times 10^{12} \mathrm{~km}}{\text{light year}} \\ & = 4.6 \times 9.461 \times 10^{9+12} \mathrm{~km} \\ & = 43.5206 \times 10^{21} \mathrm{~km} \\ & = 4.35206 \times 10^{22} \mathrm{~km} \end{aligned} \] \end{solutionordottedlines} \end{questions} \end{exercisebox}