 \subsection{Fractional Indices}                                                                                
 \begin{questions}                                                                                              
     \Question[3] Evaluate:                                                                                     
     \begin{multicols}{3}                                                                                       
         \begin{parts}                                                                                          
             \part \(\sqrt[4]{81}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \sqrt[4]{81} = 3 \)                                                                         
             \end{solutionordottedlines}                                                                        
             \part \(\sqrt[3]{64}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \sqrt[3]{64} = 4 \)                                                                         
             \end{solutionordottedlines}                                                                        
             \part \(\sqrt[5]{2^{10}}\)                                                                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \sqrt[5]{2^{10}} = 2^{2} = 4 \)                                                             
             \end{solutionordottedlines}                                                                        
         \end{parts}                                                                                            
     \end{multicols}                                                                                            
     \Question[2] Write using fractional indices. Evaluate, correct to 4 decimal places.                        
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(\sqrt[7]{11}\)                                                                                 
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 11^{\frac{1}{7}} \approx 1.4407 \)                                                          
             \end{solutionordottedlines}                                                                        
         \part \(\sqrt[3]{2^{7}}\)                                                                              
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( (2^{7})^{\frac{1}{3}} = 2^{\frac{7}{3}} \approx 5.0397 \)                                   
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[6] Evaluate:                                                                                     
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(243^{\frac{1}{5}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 243^{\frac{1}{5}} = 3 \)                                                                    
             \end{solutionordottedlines}                                                                        
         \part \(81^{\frac{1}{4}}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 81^{\frac{1}{4}} = 3 \)                                                                     
             \end{solutionordottedlines}                                                                        
         \part \(125^{\frac{1}{3}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 125^{\frac{1}{3}} = 5 \)                                                                    
             \end{solutionordottedlines}                                                                        
         \part \(64^{\frac{1}{3}}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 64^{\frac{1}{3}} = 4 \)                                                                     
             \end{solutionordottedlines}                                                                        
         \part \(216^{\frac{1}{3}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 216^{\frac{1}{3}} = 6 \)                                                                    
             \end{solutionordottedlines}                                                                        
         \part \(49^{\frac{1}{2}}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 49^{\frac{1}{2}} = 7 \)                                                                     
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[6] Evaluate:                                                                                     
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(125^{\frac{2}{3}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 125^{\frac{2}{3}} = 25 \)                                                                   
             \end{solutionordottedlines}                                                                        
         \part \(64 \frac{5}{6}\)                                                                               
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 64^{\frac{5}{6}} \approx 32 \)                                                              
             \end{solutionordottedlines}                                                                        
         \part \(216^{\frac{2}{3}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 216^{\frac{2}{3}} = 36 \)                                                                   
             \end{solutionordottedlines}                                                                        
         \part \(243^{\overline{5}}\)                                                                           
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 243^{-\frac{1}{5}} \approx 0.4822 \)                                                        
             \end{solutionordottedlines}                                                                        
         \part \(\sqrt[5]{32^{4}}\)                                                                             
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \sqrt[5]{32^{4}} = 32^{\frac{4}{5}} = 16 \)                                                 
             \end{solutionordottedlines}                                                                        
         \part \(\sqrt[3]{2^{6}}\)                                                                              
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \sqrt[3]{2^{6}} = 2^{2} = 4 \)                                                              
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[8] Simplify:                                                                                     
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(\left(c^{12}\right)^{\frac{1}{4}}\)                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(c^{12}\right)^{\frac{1}{4}} = c^{3} \)                                                
             \end{solutionordottedlines}                                                                        
         \part \(\left(c^{10}\right)^{\frac{1}{5}}\)                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(c^{10}\right)^{\frac{1}{5}} = c^{2} \)                                                
             \end{solutionordottedlines}                                                                        
         \part \(p^{\frac{3}{4}} \times p^{\frac{2}{5}}\)                                                       
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( p^{\frac{3}{4}} \times p^{\frac{2}{5}} = p^{\frac{3}{4}+\frac{2}{5}} =                      
 p^{\frac{15}{20}+\frac{8}{20}} = p^{\frac{23}{20}} \)                                                          
             \end{solutionordottedlines}                                                                        
         \part \(q^{\frac{3}{2}} \times q^{\frac{2}{3}}\)                                                       
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( q^{\frac{3}{2}} \times q^{\frac{2}{3}} = q^{\frac{3}{2}+\frac{2}{3}} =                      
 q^{\frac{9}{6}+\frac{4}{6}} = q^{\frac{13}{6}} \)                                                              
             \end{solutionordottedlines}                                                                        
         \part \(p^{\frac{3}{4}} \div p^{\frac{2}{5}}\)                                                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( p^{\frac{3}{4}} \div p^{\frac{2}{5}} = p^{\frac{3}{4}-\frac{2}{5}} =                        
 p^{\frac{15}{20}-\frac{8}{20}} = p^{\frac{7}{20}} \)                                                           
             \end{solutionordottedlines}                                                                        
         \part \(q^{\frac{3}{2}} \div q^{\frac{2}{3}}\)                                                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( q^{\frac{3}{2}} \div q^{\frac{2}{3}} = q^{\frac{3}{2}-\frac{2}{3}} =                        
 q^{\frac{9}{6}-\frac{4}{6}} = q^{\frac{5}{6}} \)                                                               
             \end{solutionordottedlines}                                                                        
         \part \(\left(2 x^{\frac{2}{3}}\right)^{3}\)                                                           
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(2 x^{\frac{2}{3}}\right)^{3} = 2^{3} \cdot x^{2} = 8x^{2} \)                          
             \end{solutionordottedlines}                                                                        
         \part \(\left(3 y^{\frac{1}{2}}\right)^{4}\)                                                           
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(3 y^{\frac{1}{2}}\right)^{4} = 3^{4} \cdot y^{2} = 81y^{2} \)                         
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[4] Evaluate:                                                                                     
     \begin{parts}                                                                                              
         \part \(\left(\frac{64}{27}\right)^{-\frac{1}{3}}\)                                                    
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(\frac{64}{27}\right)^{-\frac{1}{3}} = \left(\frac{27}{64}\right)^{\frac{1}{3}} =      
 \frac{3}{4} \)                                                                                                 
             \end{solutionordottedlines}                                                                        
         \part \(32^{-\frac{2}{5}}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( 32^{-\frac{2}{5}} \approx 0.2795 \)                                                         
             \end{solutionordottedlines}                                                                        
         \part \(\left(\frac{16}{81}\right)^{-\frac{1}{4}}\)                                                    
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(\frac{16}{81}\right)^{-\frac{1}{4}} = \left(\frac{81}{16}\right)^{\frac{1}{4}} \approx 2.3784 \)                                                                                                      
             \end{solutionordottedlines}                                                                        
         \part \(\left(\frac{32}{243}\right)^{-\frac{1}{5}}\)                                                   
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(\frac{32}{243}\right)^{-\frac{1}{5}} = \left(\frac{243}{32}\right)^{\frac{1}{5}}      
 \approx 1.9036 \)                                                                                              
             \end{solutionordottedlines}                                                                        
     \end{parts}                                                                                                
     \Question[6] Simplify, expressing the answer with positive indices.                                        
     \begin{parts}                                                                                              
         \part \(\left(3 y^{\frac{1}{2}}\right)^{-4}\)                                                          
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(3 y^{\frac{1}{2}}\right)^{-4} = \left(\frac{1}{3}\right)^{4} \cdot y^{-2} =           
 \frac{1}{81y^{2}} \)                                                                                           
             \end{solutionordottedlines}                                                                        
         \part \(x^{\frac{1}{2}} \times x^{-\frac{3}{2}}\)                                                      
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( x^{\frac{1}{2}} \times x^{-\frac{3}{2}} = x^{\frac{1}{2}-\frac{3}{2}} = x^{-1} = \frac{1}{x} \)                                          \end{solutionordottedlines}                                                                        
         \part \(x^{\frac{3}{2}} \div x^{-\frac{1}{2}}\)                                                        
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( x^{\frac{3}{2}} \div x^{-\frac{1}{2}} = x^{\frac{3}{2}+\frac{1}{2}} = x^{2} \)              
             \end{solutionordottedlines}                                                                        
         \part \(y^{\frac{2}{3}} \div y^{-\frac{1}{3}}\)                                                        
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( y^{\frac{2}{3}} \div y^{-\frac{1}{3}} = y^{\frac{2}{3}+\frac{1}{3}} = y^{1} = y \)          
             \end{solutionordottedlines}                                                                        
         \part \(\left(27 n^{-12}\right)^{\frac{1}{3}}\)                                                        
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(27 n^{-12}\right)^{\frac{1}{3}} = 27^{\frac{1}{3}} \cdot n^{-4} = 3n^{-4} =           
 \frac{3}{n^{4}} \)                                                                                             
             \end{solutionordottedlines}                                                                        
         \part \(\left(2 x^{-\frac{2}{5}}\right)^{5}\)                                                          
             \begin{solutionordottedlines}[2cm]                                                                 
                 \( \left(2 x^{-\frac{2}{5}}\right)^{5} = 2^{5} \cdot x^{-2} = 32x^{-2} = \frac{32}{x^{2}} \)   
             \end{solutionordottedlines}                                                                        
     \end{parts}                                                                                                
 \end{questions}                                                                                                

 \subsection{Scientific Notation}                                                                               
 \begin{questions}                                                                                              
     \Question[4] Write in scientific notation.                                                                 
     \begin{multicols}{2}\begin{parts}                                                                          
         \part 26000                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.6 \times 10^{4}$                                                                            
             \end{solutionordottedlines}                                                                        
         \part 4000000000000                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                 
                 $4 \times 10^{12}$                                                                             
             \end{solutionordottedlines}                                                                        
         \part 0.00072                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                 
                 $7.2 \times 10^{-4}$                                                                           
             \end{solutionordottedlines}                                                                        
         \part 0.000000206                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.06 \times 10^{-7}$                                                                          
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[4] Write in decimal form:                                                                        
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(8.6 \times 10^{2}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 860                                                                                            
             \end{solutionordottedlines}                                                                        
         \part \(7.2 \times 10^{1}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 72                                                                                             
             \end{solutionordottedlines}                                                                        
         \part \(8.72 \times 10^{-4}\)                                                                          
             \begin{solutionordottedlines}[2cm]                                                                 
                 0.000872                                                                                       
             \end{solutionordottedlines}                                                                        
         \part \(2.6 \times 10^{-7}\)                                                                           
             \begin{solutionordottedlines}[2cm]                                                                 
                 0.00000026                                                                                     
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[6] Simplify, expressing the answer in scientific notation.                                       
     \begin{parts}                                                                                              
         \part \(\left(4 \times 10^{-2}\right)^{2} \times\left(5 \times 10^{7}\right)\)                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 $1.6 \times 10^{-4} \times 5 \times 10^{7} = 8 \times 10^{3}$                                  
             \end{solutionordottedlines}                                                                        
         \part \(\left(6 \times 10^{-3}\right) \times\left(4 \times 10^{7}\right)^{2}\)                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 $6 \times 10^{-3} \times 16 \times 10^{14} = 9.6 \times 10^{12}$                               
             \end{solutionordottedlines}                                                                        
         \part \(\frac{\left(4 \times 10^{5}\right)^{3}}{\left(8 \times 10^{4}\right)^{2}}\)                    
             \begin{solutionordottedlines}[2cm]                                                                 
                 $\frac{64 \times 10^{15}}{64 \times 10^{8}} = 10^{7}$                                          
             \end{solutionordottedlines}                                                                        
         \part \(\frac{\left(2 \times 10^{-1}\right)^{5}}{\left(4 \times 10^{-2}\right)^{3}}\)                  
             \begin{solutionordottedlines}[2cm]                                                                 
                 $\frac{32 \times 10^{-5}}{64 \times 10^{-6}} = 0.5 \times 10^{1} = 5 \times 10^{0}$            
             \end{solutionordottedlines}                                                                        
     \end{parts}                                                                                                
     \Question[2] If light travels at \(3 \times 10^{5} \mathrm{~km} / \mathrm{s}\) and our galaxy is           
 approximately 80000 light years across, how many kilometres is it across? (A light year is the distance light  
 travels in a year.)                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 $3 \times 10^{5} \mathrm{~km/s} \times 60 \times 60 \times 24 \times 365.25 \times 80000 =     
 7.5696 \times 10^{17} \mathrm{~km}$                                                                            
             \end{solutionordottedlines}                                                                        
     \Question[2] The mass of a hydrogen atom is approximately \(1.674 \times 10^{-27} \mathrm{~kg}\) and the   
 mass of an electron is approximately \(9.1 \times 10^{-31} \mathrm{~kg}\). How many electrons, correct to the  
 nearest whole number, will have the same mass as a single hydrogen atom?                                       
             \begin{solutionordottedlines}[2cm]                                                                 
                 $\frac{1.674 \times 10^{-27}}{9.1 \times 10^{-31}} \approx 1839$ electrons                     
             \end{solutionordottedlines}                                                                        
     \Question[2] In a lottery there are \(\frac{45 \times 44 \times 43 \times 42 \times 41 \times 40}{720}\)   
 different possible outcomes. If I mark each outcome on an entry form one at a time, and it takes me an average 
 of 1 minute to mark each outcome, how long will it take me to cover all different possible outcomes?           
             \begin{solutionordottedlines}[2cm]                                                                 
                 $\frac{45 \times 44 \times 43 \times 42 \times 41 \times 40}{720} = 8,145,060$ minutes         
             \end{solutionordottedlines}                                                                        
 \end{questions}                                                                                                
                                                                                                                
 \subsection{Significant Figures}                                                                               
                                                                                                                
 \begin{questions}                                                                                              
     \Question[4] Write in scientific notation, correct to 2 significant figures.                               
     \begin{multicols}{2}\begin{parts}                                                                          
         \part 368.2                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 $3.7 \times 10^{2}$                                                                            
             \end{solutionordottedlines}                                                                        
         \part 278000                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.8 \times 10^{5}$                                                                            
             \end{solutionordottedlines}                                                                        
         \part 0.004321                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                 
                 $4.3 \times 10^{-3}$                                                                           
             \end{solutionordottedlines}                                                                        
         \part 0.000021906                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.2 \times 10^{-5}$                                                                           
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[4] Use a calculator to evaluate the following, giving the answer in scientific notation correct  
 3 significant figures.                                                                                         
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(3.24 \times 0.067\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.17 \times 10^{-1}$                                                                          
             \end{solutionordottedlines}                                                                        
         \part \(4.736 \times 10^{13} \times 2.34 \times 10^{-6}\)                                              
             \begin{solutionordottedlines}[2cm]                                                                 
                 $1.11 \times 10^{8}$                                                                           
             \end{solutionordottedlines}                                                                        
         \part \(0.0276^{2} \times \sqrt{0.723}\)                                                               
             \begin{solutionordottedlines}[2cm]                                                                 
                 $5.48 \times 10^{-4}$                                                                          
             \end{solutionordottedlines}                                                                        
         \part \(\frac{6.54\left(5.26^{2}+3.24\right)}{5.4+\sqrt{6.34}}\)                                       
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.97 \times 10^{1}$                                                                           
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
     \Question[4] Use a calculator to evaluate, giving the answer in scientific notation correct to 4 significa 
 figures.                                                                                                       
     \begin{multicols}{2}\begin{parts}                                                                          
         \part \(1.234 \times 0.1988\)                                                                          
             \begin{solutionordottedlines}[2cm]                                                                 
                 $2.453 \times 10^{-1}$                                                                         
             \end{solutionordottedlines}                                                                        
         \part \(1.234 \div 0.1988\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                 
                 $6.206 \times 10^{0}$                                                                          
             \end{solutionordottedlines}                                                                        
         \part \(1.9346^{3}\)                                                                                   
             \begin{solutionordottedlines}[2cm]                                                                 
                 $7.238 \times 10^{0}$                                                                          
             \end{solutionordottedlines}                                                                        
         \part \(\left(7.919 \times 10^{21}\right)^{2}\)                                                        
             \begin{solutionordottedlines}[2cm]                                                                 
                 $6.271 \times 10^{43}$                                                                         
             \end{solutionordottedlines}                                                                        
     \end{parts}\end{multicols}                                                                                 
 \end{questions}                                                                                                
