                                                                                                                                                            
 \begin{examplebox}                                                                                                                                         
     \subsection{Examples:}                                                                                                                                 
     \begin{questions}                                                                                                                                      
         \Question[1] Express as a power or as a product of powers.                                                                                         
         \begin{parts}                                                                                                                                      
             \part \(5 \times 5 \times 5\)                                                                                                                  
             \begin{solutionordottedlines}                                                                                                                  
                 \(5 \times 5 \times 5=5^{3} \quad\)                                                                                                        
             \end{solutionordottedlines}                                                                                                                    
             \part \(3 \times 3 \times 7 \times 7 \times 7 \times 7\)                                                                                       
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 \times 3 \times 7 \times 7 \times 7 \times 7=3^{2} \times 7^{4}\)                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part                                                                                                                                          
         \end{parts}                                                                                                                                        
         \Question[1] Express each number as a power of a prime.                                                                                            
         \begin{parts}                                                                                                                                      
             \part 81                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(81=3^{4}\)                                                                                                                               
             \end{solutionordottedlines}                                                                                                                    
             \part 128                                                                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(128=2^{7}\)                                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
         \end{parts}                                                                                                                                        
     \end{questions}                                                                                                                                        
 \end{examplebox}                                                                                                                                           
                                                                                                                                                            
 Now some definitions:                                                                                                                                      
 \begin{boxdef}                                                                                                                                             
     \textbf{Base:}                                                                                                                                         
     \begin{solutionordottedlines}[1cm]                                                                                                                     
         The number 2 in \(2^{4}\) is called the base.                                                                                                      
     \end{solutionordottedlines}                                                                                                                            
 \end{boxdef}                                                                                                                                               
 \begin{boxdef}                                                                                                                                             
     \textbf{Index:}                                                                                                                                        
     \begin{solutionordottedlines}[1cm]                                                                                                                     
         The number 4 in \(2^{4}\) is called the index or exponent.                                                                                         
     \end{solutionordottedlines}                                                                                                                            
 \end{boxdef}                                                                                                                                               
                                                                                                                                                            
 \begin{boxlaw}                                                                                                                                             
     \subsection*{Index Laws}                                                                                                                               
     \subsubsection*{Product of Powers}                                                                                                                     
         To multiply powers of the same base, add the indices.                                                                                              
                                                                                                                                                            
         \[                                                                                                                                                 
         a^{m} a^{n}=a^{m+n}                                                                                                                                
         \]                                                                                                                                                 
                                                                                                                                                            
     \subsubsection*{Division of Powers}                                                                                                                    
         To divide powers of the same base, subtract the indices.                                                                                           
                                                                                                                                                            
         \[                                                                                                                                                 
         \frac{a^{m}}{a^{n}}=a^{m-n} \quad \text { where } m>n \text { and } a \neq 0                                                                       
         \]                                                                                                                                                 
     \subsubsection*{Powers of Powers}                                                                                                                      
         To raise a power to a power, multiply the indices.                                                                                                 
                                                                                                                                                            
         \[                                                                                                                                                 
         \left(a^{m}\right)^{n}=a^{m n}                                                                                                                     
         \]                                                                                                                                                 
     \subsubsection*{Power of Products}                                                                                                                     
                                                                                                                                                            
         A power of a product is the product of the powers.                                                                                                 
                                                                                                                                                            
         \[(a b)^{m}=a^{m} b^{m}\]                                                                                                                          
                                                                                                                                                            
     \subsubsection*{Power of Quotient}                                                                                                                     
         A power of a quotient is the quotient of the powers.                                                                                               
                                                                                                                                                            
         \[                                                                                                                                                 
         \left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}} \quad \text { where } b \neq 0                                                                    
         \]                                                                                                                                                 
     \subsubsection*{Zero Exponent Law}                                                                                                                     
         Any base raised to the power of $0$ is equal to $1$.                                                                                               
         \[a^0 = 1\]                                                                                                                                        
     \subsubsection*{Negative Exponent Law (next lesson)}                                                                                                   
          \[ a^{-n} = \frac{1}{a^n} \]                                                                                                                      
          A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.                                           
 \end{boxlaw}                                                                                                                                               
                                                                                                                                                            
 Now with these laws in mind, try the following 10 examples:                                                                                                
 \begin{examplebox}                                                                                                                                         
     \subsection{Examples}                                                                                                                                  
     \begin{questions}                                                                                                                                      
         \Question[10] Simplify, expressing the answer in index form.                                                                                       
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(3^{2} \times 3^{4}\)                                                                                                                   
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(3^{2} \times 3^{4}=3^{6}\)                                                                                                           
                 \end{solutionordottedlines}                                                                                                                
             \part \(a^{3} \times a^{5}\)                                                                                                                   
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(a^{3} \times a^{5}=a^{8}\)                                                                                                           
                 \end{solutionordottedlines}                                                                                                                
             \part \(3 x^{2} \times x^{3}\)                                                                                                                 
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(3 x^{2} \times x^{3}=3 x^{5}\)                                                                                                       
                 \end{solutionordottedlines}                                                                                                                
             \part \(2 a^{2} b^{3} \times 5 a b^{2}\)                                                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(2 a^{2} b^{3} \times 5 a b^{2}=10 a^{3} b^{5}\)                                                                                      
                 \end{solutionordottedlines}                                                                                                                
             \part \(\frac{3^{5}}{3^{2}}\)                                                                                                                  
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(\frac{3^{5}}{3^{2}}=3^{3}\)                                                                                                          
                 \end{solutionordottedlines}                                                                                                                
             \part \(\frac{9^{5}}{9^{4}}\)                                                                                                                  
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(\frac{9^{5}}{9^{4}}=9^{1}\)                                                                                                          
                 \end{solutionordottedlines}                                                                                                                
             \part \(10^{6} \div 10^{4}\)                                                                                                                   
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(10^{6} \div 10^{4}=10^{2}\)                                                                                                          
                 \end{solutionordottedlines}                                                                                                                
             \part \(a^{7} \div a^{4}\)                                                                                                                     
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(a^{7} \div a^{4}=a^{3}\)                                                                                                             
                 \end{solutionordottedlines}                                                                                                                
             \part \(\frac{3 y^{4}}{y}\)                                                                                                                    
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(\frac{3 y^{4}}{y}=3 y^{3}\)                                                                                                          
                 \end{solutionordottedlines}                                                                                                                
             \part \(\frac{6 x^{5}}{2 x^{3}}\)                                                                                                              
                 \begin{solutionordottedlines}[2cm]                                                                                                         
                     \(\frac{6 x^{5}}{2 x^{3}}=3 x^{2}\)                                                                                                    
                 \end{solutionordottedlines}                                                                                                                
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Double marks for each of these:                                                                                                       
         \begin{parts}                                                                                                                                      
             \part \[\frac{3 x^{3} y^{2}}{4 x y} \times \frac{6 x^{2} y^{3}}{x^{3} y^{2}}\]                                                                 
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{3 x^{3} y^{2}}{4 x y} \times \frac{6 x^{2} y^{3}}{x^{3} y^{2}}=\frac{18 x^{5} y^{5}}{4 x^{4} y^{3}}=\frac{9 x y^{2}}{2}\)          
             \end{solutionordottedlines}                                                                                                                    
             \part \[\frac{8 a^{2} b^{3}}{3 a^{3} b} \div \frac{4 a b^{2}}{9 a^{3} b^{5}}\]                                                                 
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{8 a^{2} b^{3}}{3 a^{3} b} \div \frac{4 a b^{2}}{9 a^{3} b^{5}}=\frac{8 a^{2} b^{3}}{3 a^{3} b} \times \frac{9 a^{3} b^{5}}{4 a     
 b^{2}}=6 a b^{5}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                    
         \end{parts}                                                                                                                                        
         \Question[6] And a $1.5\times$ multiplier for these:                                                                                               
         \begin{parts}                                                                                                                                      
             \part \(\left(\frac{2}{3}\right)^{2}\)                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{2}{3}\right)^{2}=\frac{2^{2}}{3^{2}}=\frac{4}{9}\)                                                                           
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{m}{n}\right)^{5}\)                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{m}{n}\right)^{5}=\frac{m^{5}}{n^{5}}\)                                                                                       
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{x^{3}}{y^{2}}\right)^{2} \times\left(\frac{y}{x}\right)^{4}\)                                                              
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{x^{3}}{y^{2}}\right)^{2} \times\left(\frac{y}{x}\right)^{4}=\frac{x^{6}}{y^{4}} \times \frac{y^{4}}{x^{4}}=x^{2}\)           
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{2 x^{2}}{3}\right)^{2} \div \frac{4 x^{3}}{9}\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{2 x^{2}}{3}\right)^{2} \div \frac{4 x^{3}}{9}=\frac{4 x^{4}}{9} \times \frac{9}{4 x^{3}}=x\)                                 
             \end{solutionordottedlines}                                                                                                                    
         \end{parts}                                                                                                                                        
         \begin{parts}                                                                                                                                      
             \part \(\left(5 a^{3}\right)^{0}\)                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(5 a^{3}\right)^{0}=1\)                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{6 x^{2} y}{x y^{2}} \times \frac{y^{3} x}{2 y^{2} x^{2}}\)                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{6 x^{2} y}{x y^{2}} \times \frac{y^{3} x}{2 y^{2} x^{2}}=3\)                                                                       
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(m n^{2}\right)^{0}\)                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(m n^{2}\right)^{0}=1\)                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(a^{4} b^{2}\right)^{3}\)                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(a^{4} b^{2}\right)^{3}=a^{12} b^{6}\)                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(2 a^{4}\right)^{3}\)                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(2 a^{4}\right)^{3}=2^{3} a^{12}=8 a^{12}\)                                                                                         
             \end{solutionordottedlines}                                                                                                                    
             \part \(2\left(x^{2} y\right)^{0} \times\left(x^{2} y^{3}\right)^{3}\)                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2\left(x^{2} y\right)^{0} \times\left(x^{2} y^{3}\right)^{3}=2 x^{6} y^{9}\)                                                             
             \end{solutionordottedlines}                                                                                                                    
             \part                                                                                                                                          
         \end{parts}                                                                                                                                        
     \end{questions}                                                                                                                                        
 \end{examplebox}                                                                                                                                           
                                                                                                                                                            
 \fbox{\fbox{\parbox{\textwidth}{\textbf{Note: }There are different possible interpretations of the word 'simplify'. There may be more than one acceptable  
 simplified form.                                                                                                                                           
 }}}                                                                                                                                                        
                                                                                                                                                            
                                                                                                                                                            
 \begin{exercisebox}                                                                                                                                        
     \subsection{Exercises:}                                                                                                                                
     \begin{questions}                                                                                                                                      
         \Question[3] State the base and index of:                                                                                                          
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(6^{4}=\fillin[Base: 6, Index: 4]\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
             \part \(7^{3}=\fillin[Base: 7, Index: 3]\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
             \part \(8^{2}=\fillin[Base: 8, Index: 2]\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Express as a power of a prime number.                                                                                                 
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part $8 = \fillin[2^{3}]$                                                                                                                     
             \part $27 = \fillin[3^{3}]$                                                                                                                    
             \part $64 = \fillin[2^{6}]$                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Evaluate:                                                                                                                             
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(3^{4}=\fillin[81]\)                                                                                                                    
             \part \(2^{7}=\fillin[128]\)                                                                                                                   
             \part \(5^{5}=\fillin[3125]\)                                                                                                                  
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Express as a product of powers of prime numbers.                                                                                      
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part $18=\fillin[2 \times 3^{2}]$                                                                                                             
             \part $24=\fillin[2^{3} \times 3]$                                                                                                             
             \part $144=\fillin[2^{4} \times 3^{2}]$                                                                                                        
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Simplify:                                                                                                                             
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(2^{7} \times 2^{3}\)                                                                                                                   
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2^{7} \times 2^{3}=2^{10}\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(3^{3} \times 3^{4} \times 3^{5}\)                                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(3^{3} \times 3^{4} \times 3^{5}=3^{12}\)                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(3 x^{2} \times 4 x^{3}\)                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(3 x^{2} \times 4 x^{3}=12 x^{5}\)                                                                                                        
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Simplify:                                                                                                                             
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(a^{2} b^{3} \times b^{2}\)                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(a^{2} b^{3} \times b^{2}=a^{2} b^{5}\)                                                                                                   
             \end{solutionordottedlines}                                                                                                                    
             \part \(a^{3} b \times a^{2} b^{3}\)                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(a^{3} b \times a^{2} b^{3}=a^{5} b^{4}\)                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(2 x y^{2} \times 3 x^{2} y\)                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2 x y^{2} \times 3 x^{2} y=6 x^{3} y^{3}\)                                                                                               
             \end{solutionordottedlines}                                                                                                                    
             \part \(4 a^{3} b^{2} \times a^{2} b^{4}\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(4 a^{3} b^{2} \times a^{2} b^{4}=4 a^{5} b^{6}\)                                                                                         
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[6]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{3^7}{3^2}\)                                                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{3^7}{3^2}=3^{5}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{2^{6}}{2^{2}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{2^{6}}{2^{2}}=2^{4}\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(10^{7} \div 10^{2}\)                                                                                                                   
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(10^{7} \div 10^{2}=10^{5}\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{10^{12}}{10^{4}}\)                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{10^{12}}{10^{4}}=10^{8}\)                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{2 x^{3}}{x^{2}}\)                                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{2 x^{3}}{x^{2}}=2 x\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{6 x^{5}}{2 x^{2}}\)                                                                                                              
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{6 x^{5}}{2 x^{2}}=3 x^{3}\)                                                                                                        
             \end{solution                                                                                                                                  
                                                                                                                                                            