\begin{problembox}                                                                                                                                                             
     \begin{questions}                                                                                                                                                          
         \Question[4] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(\frac{4 m^{2} n^{5} p^{-6}}{16 m^{-2} n^{5} p^{3}}\)                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{4}{16} \cdot m^{2-(-2)} \cdot n^{5-5} \cdot p^{-6-3}\) \\                                                                                             
                 \(=\frac{1}{4} \cdot m^{4} \cdot p^{-9}\) \\                                                                                                                   
                 \(=\frac{m^{4}}{4p^{9}}\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(2^{2} y^{3}\right)^{-5}\)                                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(2^{2 \cdot -5}) \cdot (y^{3 \cdot -5})\) \\                                                                                                                
                 \(=2^{-10} \cdot y^{-15}\) \\                                                                                                                                  
                 \(=\frac{1}{2^{10} y^{15}}\)                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(5^{-2} x^{3}\right)^{-5}\)                                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(5^{-2 \cdot -5}) \cdot (x^{3 \cdot -5})\) \\                                                                                                               
                 \(=5^{10} \cdot x^{-15}\) \\                                                                                                                                   
                 \(=\frac{5^{10}}{x^{15}}\)                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(3^{-3} a^{2} b^{-1}\right)^{-4}\)                                                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(3^{-3 \cdot -4}) \cdot (a^{2 \cdot -4}) \cdot (b^{-1 \cdot -4})\) \\                                                                                       
                 \(=3^{12} \cdot a^{-8} \cdot b^{4}\) \\                                                                                                                        
                 \(=\frac{3^{12} b^{4}}{a^{8}}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(4 a^{2} \times 5 a^{-3}\)                                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4 \cdot 5 \cdot a^{2-3}\) \\                                                                                                                                
                 \(=20 a^{-1}\) \\                                                                                                                                              
                 \(=\frac{20}{a}\)                                                                                                                                              
             \end{solutionordottedlines}                                                                                                                                        
             \part \(14 a^{-4} \div 7 a^{-5}\)                                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{14}{7} \cdot a^{-4-(-5)}\) \\                                                                                                                         
                 \(=2 a^{1}\) \\                                                                                                                                                
                 \(=2a\)                                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\frac{2 m^{3} n^{4}}{(5 m)^{2}} \times \frac{10 m}{3 n^{-4}}\)                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{2 m^{3} n^{4}}{25 m^{2}} \cdot \frac{10 m}{3 n^{-4}}\) \\                                                                                             
                 \(=\frac{2}{25} \cdot \frac{10}{3} \cdot m^{3-2+1} \cdot n^{4+4}\) \\                                                                                          
                 \(=\frac{20}{75} \cdot m^{2} \cdot n^{8}\) \\                                                                                                                  
                 \(=\frac{4}{15} m^{2} n^{8}\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Evaluate:                                                                                                                                                 
         \begin{parts}                                                                                                                                                          
             \part \(49^{\frac{1}{2}}\)                                                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\sqrt{49}\) \\                                                                                                                                              
                 \(=7\)                                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part \(125^{\frac{2}{3}}\)                                                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\sqrt[3]{125^{2}}\) \\                                                                                                                                      
                 \(=\sqrt[3]{15625}\) \\                                                                                                                                        
                 \(=25\)                                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(\frac{1}{8}\right)^{-\frac{2}{3}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(8^{\frac{2}{3}})\) \\                                                                                                                                      
                 \(=\sqrt[3]{8^{2}}\) \\                                                                                                                                        
                 \(=\sqrt[3]{64}\) \\                                                                                                                                           
                 \(=4\)                                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[4] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(3 b^{\frac{2}{3}} \times 4 b\)                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3 \cdot 4 \cdot b^{\frac{2}{3}+1}\) \\                                                                                                                      
                 \(=12 b^{\frac{5}{3}}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(p^{\frac{2}{3}} \div p^{\frac{1}{2}}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=p^{\frac{2}{3}-\frac{1}{2}}\) \\                                                                                                                            
                 \(=p^{\frac{4}{6}-\frac{3}{6}}\) \\                                                                                                                            
                 \(=p^{\frac{1}{6}}\)                                                                                                                                           
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(2 x^{-\frac{1}{3}}\right)^{-2}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=2^{-2} \cdot x^{\frac{2}{3}}\) \\                                                                                                                           
                 \(=\frac{1}{4} x^{\frac{2}{3}}\)                                                                                                                               
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(8 p^{-2} q^{3}\right)^{\frac{1}{2}}\)                                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=8^{\frac{1}{2}} \cdot p^{-1} \cdot q^{\frac{3}{2}}\) \\                                                                                                     
                 \(=2 \cdot \frac{q^{\frac{3}{2}}}{p}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Write in scientific notation.                                                                                                                             
         \begin{parts}                                                                                                                                                          
             \part 164000000                                                                                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.64 \times 10^{8}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.0047                                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.7 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.0035                                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3.5 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Write in decimal form.                                                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(6.8 \times 10^{4}\)                                                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=68000\)                                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(9.4 \times 10^{-2}\)                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=0.094\)                                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(3.2 \times 10^{-4}\)                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=0.00032\)                                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[2] Simplify, writing each answer in scientific notation.                                                                                                     
         \begin{parts}                                                                                                                                                          
             \part \(\left(3.1 \times 10^{4}\right) \times\left(2 \times 10^{-2}\right)\)                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3.1 \cdot 2 \times 10^{4-2}\) \\                                                                                                                            
                 \(=6.2 \times 10^{2}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\frac{\left(3 \times 10^{4}\right)^{3}}{9 \times 10^{-2}}\)                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{3^{3} \times 10^{12}}{9 \times 10^{-2}}\) \\                                                                                                          
                 \(=\frac{27}{9} \times 10^{12+2}\) \\                                                                                                                          
                 \(=3 \times 10^{14}\)                                                                                                                                          
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[6] Write in scientific notation correct to the number of significant figures indicated in the brackets.                                                      
         \begin{multicols}{2}\begin{parts}                                                                                                                                      
             \part 18.62\hfill{}(2)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.9 \times 10^{1}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part 18.62\hfill{}(3)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.86 \times 10^{1}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 18.62\hfill{}(1)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=2 \times 10^{1}\)                                                                                                                                           
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.004276\hfill{}(2)                                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.3 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 5973.4\hfill{}(2)                                                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=6.0 \times 10^{3}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.473952\hfill{}(3)                                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.74 \times 10^{-1}\)                                                                                                                                       
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}\end{multicols}                                                                                                                                             
     \end{questions}                                                                                                                                                            
 \end{problembox}