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}                                                                                              



