This section is tricky as negative numbers always tend to be. The most important thing you will learn here is:

\begin{lawbox}
    \large
    \[a^{-n} = \frac{1}{a^n}\]
\end{lawbox}

I want you to take a second to realise that this \fuzzyfont{breaks} your interpretation of $a$ being raised to some power $n$ as being $n$ successive multiplications of the number $a$.
\[a^n = \underbrace{a \times a \times a \ldots}_{n} \]

Suddenly what does it mean to multiply $a$ by itself \textit{$-n$} times?!

Here is some working out space for you to reconcile with this fact.
\begin{solutionorbox}[2in]
 \textbf{Proof:}

 Consider the product of \( a^n \) and \( a^{-n} \):
 \[ a^n \cdot a^{-n} = a^{n + (-n)} = a^0 \]

 Since anything raised to the power of 0 is 1, we have:
 \[ a^0 = 1 \]

 Therefore:
 \[ a^n \cdot a^{-n} = 1 \]

 Dividing both sides by \( a^n \), we get:
 \[ \frac{a^n \cdot a^{-n}}{a^n} = \frac{1}{a^n} \]

 Simplifying the left side, we have:
 \[ a^{-n} = \frac{1}{a^n} \]

 Hence proved.
\end{solutionorbox}

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \item[Worked Example:] \[2^{-3} = \frac{1}{2^3} = \frac{1}{8}\]
        \Question[4] Evaluate
        \begin{parts}\begin{multicols}{2}
            \part \(6^{-2}\)
            \begin{solutionordottedlines}[2cm]
                \(6^{-2}=\frac{1}{6^{2}}\)
                \(=\frac{1}{36}\)
            \end{solutionordottedlines}
            \part \(4^{-3}\)
            \begin{solutionordottedlines}[2cm]
                \(4^{-3}=\frac{1}{4^{3}}\)
                \(=\frac{1}{64}\)
            \end{solutionordottedlines}
            \part \(2^{-7}\)
            \begin{solutionordottedlines}[2cm]
                \(2^{-7}=\frac{1}{2^{7}}\)
                \(=\frac{1}{128}\)
            \end{solutionordottedlines}
            \part \(10^{-3}\)
            \begin{solutionordottedlines}[2cm]
                \(10^{-3}=\frac{1}{10^{3}}\)
                \(=\frac{1}{1000}\)
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \Question[3] These ones just reciprocate!
        \begin{parts}\begin{multicols}{3}
            \part \(\left(\frac{1}{3}\right)^{-1}\)
            \begin{solutionordottedlines}[3cm]
                \(\left(\frac{1}{3}\right)^{-1}=\frac{3}{1}\)
                \(=3\)
            \end{solutionordottedlines}
            \part \(\left(\frac{2}{7}\right)^{-2}\)
            \begin{solutionordottedlines}[3cm]
                \(\left(\frac{2}{7}\right)^{-2}=\left(\frac{7}{2}\right)^{2}\)
                \(=\frac{49}{4}\)
            \end{solutionordottedlines}
            \part \(\left(4 \frac{1}{4}\right)^{-2}\)
            \begin{solutionordottedlines}[3cm]
                \(\left(4 \frac{1}{4}\right)^{-2}=\left(\frac{17}{4}\right)^{-2}\)
                \(=\left(\frac{4}{17}\right)^{2}\)
                \(=\frac{16}{289}\)
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \Question[8] Write as a single power and then evaluate:
        \begin{parts}
            \part \(3^{4} \times 3^{-2}\)
            \begin{solutionordottedlines}[2cm]
                \(3^{4} \times 3^{-2}=3^{2}\)
            \end{solutionordottedlines}
            \part \(5^{7} \times 5^{-8}\)
            \begin{solutionordottedlines}[2cm]
                \(5^{7} \times 5^{-8}=5^{-1}\)

                \[
                =9
                \]
            \end{solutionordottedlines}
            \part \(13^{-8} \times 13^{15} \times 13^{-7}\)
            \begin{solutionordottedlines}[2cm]
                \(\begin{aligned} 13^{-8} \times 13^{15} \times 13^{-7} & =13^{0} \\ & =1\end{aligned}\)

                \[
                =1
                \]

                \[
                =\frac{1}{5}
                \]
            \end{solutionordottedlines}
            \part \(\left(\frac{2}{3}\right)^{-6} \times\left(\frac{2}{3}\right)^{4}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(\frac{2}{3}\right)^{-6} \times\left(\frac{2}{3}\right)^{4}=\left(\frac{2}{3}\right)^{-2}\)

                \[
                =\left(\frac{3}{2}\right)^{2}
                \]

                \[
                =\frac{9}{4}
                \]
            \end{solutionordottedlines}
            \begin{multicols}{2}
            \part \[\frac{2^{4}}{2^{5}}\]
            \begin{solutionordottedlines}[4cm]
                \(\frac{2^{4}}{2^{5}}=2^{-1}\)
                \(=\frac{1}{2}\)
            \end{solutionordottedlines}
            \part \[\frac{3^{4}}{3^{7}}\]
            \begin{solutionordottedlines}[4cm]
                \(\frac{3^{4}}{3^{7}}=3^{-3}\)
                \(=\frac{1}{3^{3}}\)
                \(=\frac{1}{27}\)
            \end{solutionordottedlines}
            \part \[\frac{5}{5^{3}}\]
            \begin{solutionordottedlines}[4cm]
                \(\frac{5}{5^{3}}=5^{-2}\)
                \(=\frac{1}{5^{2}}\)
                \(=\frac{1}{25}\)
            \end{solutionordottedlines}
            \part \[\frac{3^{4}}{3^{6}}\]
            \begin{solutionordottedlines}[4cm]
                \(\frac{3^{4}}{3^{6}}=3^{-2}\)
                \(=\frac{1}{3^{2}}\)
                \(=\frac{1}{9}\)
            \end{solutionordottedlines}
            \end{multicols}
        \end{parts}
        \Question[4] Simplify, expressing the answers with positive indices
        \begin{parts}\begin{multicols}{2}
            \part \(a^{2} b^{-3} \times a^{-4} b^{5}\)
            \begin{solutionordottedlines}[3cm]
                \(a^{2} b^{-3} \times a^{-4} b^{5}=a^{2-4} \times b^{-3+5}\)
                \(=\frac{1}{a^{2}} \times b^{2}\)
                \(=\frac{b^{2}}{a^{2}}\)
            \end{solutionordottedlines}
            \part \(\frac{x^{2} y^{3}}{x^{3} y^{2}}\)
            \begin{solutionordottedlines}[3cm]
                \(\frac{x^{2} y^{3}}{x^{3} y^{2}}=x^{-1} y^{1}\)
                \(=\frac{1}{x} \times y\)
                \(=\frac{y}{x}\)
            \end{solutionordottedlines}
            \part \(\left(2 a^{-2} b^{3}\right)^{-2}\)
            \begin{solutionordottedlines}[3cm]
                \(\left(2 a^{-2} b^{3}\right)^{-2}=2^{-2} \times a^{4} \times b^{-6}\)
                \(=\frac{1}{2^{2}} \times a^{4} \times \frac{1}{b^{6}}\)
                \(=\frac{a^{4}}{4 b^{6}}\)
            \end{solutionordottedlines}
            \part \(\left(\frac{3 m^{2}}{n}\right)^{-4}\)
            \begin{solutionordottedlines}[3cm]
                \(\left(\frac{3 m^{2}}{n}\right)^{-4}=\left(\frac{n}{3 m^{2}}\right)^{4}\)
                \(=\frac{n^{4}}{81 m^{8}}\)
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
    \end{questions}
\end{examplebox}

\fbox{\fbox{\parbox{\textwidth}{Note that in general \(\left(\frac{a}{b}\right)^{-1}=\frac{b}{a}\).
}}}


 \begin{exercisebox}                                                                                                                                        
     \subsection{Exercises:}                                                                                                                                
     \begin{questions}                                                                                                                                      
         \Question[4] Express with a positive index and then evaluate.                                                                                      
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(2^{-1}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2^{-1} = \frac{1}{2}\)\\                                                                                                                 
                 \(2^{-1} = 0.5\)                                                                                                                           
             \end{solutionordottedlines}                                                                                                                    
             \part \(5^{-1}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(5^{-1} = \frac{1}{5}\)\\                                                                                                                 
                 \(5^{-1} = 0.2\)                                                                                                                           
             \end{solutionordottedlines}                                                                                                                    
             \part \(2^{-4}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2^{-4} = \frac{1}{2^4}\)\\                                                                                                               
                 \(2^{-4} = 0.0625\)                                                                                                                        
             \end{solutionordottedlines}                                                                                                                    
             \part \(3^{-3}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(3^{-3} = \frac{1}{3^3}\)\\                                                                                                               
                 \(3^{-3} = \frac{1}{27}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Write each fraction as a power of a prime with a negative index.                                                                      
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{1}{8}\)                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{1}{8} = 2^{-3}\)                                                                                                                   
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{1}{9}\)                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{1}{9} = 3^{-2}\)                                                                                                                   
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{1}{16}\)                                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{1}{16} = 2^{-4}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{1}{64}\)                                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{1}{64} = 2^{-6}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Express with positive indices, evaluating where possible.                                                                             
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(a^{-3}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(a^{-3} = \frac{1}{a^3}\)                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(x^{-7}\)                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(x^{-7} = \frac{1}{x^7}\)                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{3}{a^{-4}}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{3}{a^{-4}} = 3a^4\)                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{5}{x^{-5}}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{5}{x^{-5}} = 5x^5\)                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Simplify and then evaluate                                                                                                            
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(\frac{1}{4}\right)^{-1}\)                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{1}{4}\right)^{-1} = 4\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{2}{5}\right)^{-2}\)                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{2}{5}\right)^{-2} = \left(\frac{5}{2}\right)^2\)\\                                                                           
                 \(\left(\frac{2}{5}\right)^{-2} = \frac{25}{4}\)                                                                                           
             \end{solutionordottedlines}                                                                                                                    
             \part \(3^{5} \times 3^{-2}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(3^{5} \times 3^{-2} = 3^{5-2}\)\\                                                                                                        
                 \(3^{5} \times 3^{-2} = 3^3\)\\                                                                                                            
                 \(3^{5} \times 3^{-2} = 27\)                                                                                                               
             \end{solutionordottedlines}                                                                                                                    
             \part \(5^{11} \times 5^{-8}\)                                                                                                                 
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(5^{11} \times 5^{-8} = 5^{11-8}\)\\                                                                                                      
                 \(5^{11} \times 5^{-8} = 5^3\)\\                                                                                                           
                 \(5^{11} \times 5^{-8} = 125\)                                                                                                             
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{2^{3}}{2^{6}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{2^{3}}{2^{6}} = 2^{3-6}\)\\                                                                                                        
                 \(\frac{2^{3}}{2^{6}} = 2^{-3}\)\\                                                                                                         
                 \(\frac{2^{3}}{2^{6}} = \frac{1}{8}\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{4^{2}}{4^{4}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{4^{2}}{4^{4}} = 4^{2-4}\)\\                                                                                                        
                 \(\frac{4^{2}}{4^{4}} = 4^{-2}\)\\                                                                                                         
                 \(\frac{4^{2}}{4^{4}} = \frac{1}{16}\)                                                                                                     
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{8^{6}}{8^{7}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{8^{6}}{8^{7}} = 8^{6-7}\)\\                                                                                                        
                 \(\frac{8^{6}}{8^{7}} = 8^{-1}\)\\                                                                                                         
                 \(\frac{8^{6}}{8^{7}} = \frac{1}{8}\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{20^{4}}{20^{6}}\)                                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{20^{4}}{20^{6}} = 20^{4-6}\)\\                                                                                                     
                 \(\frac{20^{4}}{20^{6}} = 20^{-2}\)\\                                                                                                      
                 \(\frac{20^{4}}{20^{6}} = \frac{1}{400}\)                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Express with negative index.                                                                                                          
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(\frac{3}{x}\)                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{3}{x} = 3x^{-1}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{5}{x^{2}}\)                                                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{5}{x^{2}} = 5x^{-2}\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{8}{x^{4}}\)                                                                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{8}{x^{4}} = 8x^{-4}\)                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Evaluate.                                                                                                                             
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(\left(\frac{1}{2}\right)^{-1}\)                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{1}{2}\right)^{-1} = 2\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{2}{3}\right)^{-1}\)                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}\)                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{1}{2}\right)^{-2}\)                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(\frac{1}{2}\right)^{-2} = 2^2\)\\                                                                                                  
                 \(\left(\frac{1}{2}\right)^{-2} = 4\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Simplify, expressing the answer with positive indices.                                                                                
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(x^{-6} y^{4} \times x^{2} y^{-2}\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(x^{-6} y^{4} \times x^{2} y^{-2} = x^{-6+2} y^{4-2}\)\\                                                                                  
                 \(x^{-6} y^{4} \times x^{2} y^{-2} = x^{-4} y^{2}\)\\                                                                                      
                 \(x^{-6} y^{4} \times x^{2} y^{-2} = \frac{y^2}{x^4}\)                                                                                     
             \end{solutionordottedlines}                                                                                                                    
             \part \(2 a^{-1} b^{5} \times 7 a b^{-3}\)                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(2 a^{-1} b^{5} \times 7 a b^{-3} = 14 a^{-1+1} b^{5-3}\)\\                                                                               
                 \(2 a^{-1} b^{5} \times 7 a b^{-3} = 14 b^{2}\)                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{8 a^{-4}}{2 a^{6}}\)                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{8 a^{-4}}{2 a^{6}} = 4 a^{-4-6}\)\\                                                                                                
                 \(\frac{8 a^{-4}}{2 a^{6}} = 4 a^{-10}\)\\                                                                                                 
                 \(\frac{8 a^{-4}}{2 a^{6}} = \frac{4}{a^{10}}\)                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{36 h^{-9}}{9 h^{-4}}\)                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{36 h^{-9}}{9 h^{-4}} = 4 h^{-9+4}\)\\                                                                                              
                 \(\frac{36 h^{-9}}{9 h^{-4}} = 4 h^{-5}\)\\                                                                                                
                 \(\frac{36 h^{-9}}{9 h^{-4}} = \frac{4}{h^{5}}\)                                                                                           
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[6] Fill in the missing term                                                                                                              
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(6^{4} \times \ldots=6^{2}\)                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(6^{4} \times 6^{-2} = 6^{2}\)                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(m^{5} \times \ldots=m^{-6}\)                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(m^{5} \times m^{-11} = m^{-6}\)                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(d^{-7} \div \ldots=d^{15}\)                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(d^{-7} \div d^{-22} = d^{15}\)                                                                                                           
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(a^{5}\right) \cdots=a^{-15}\)                                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(a^{5}\right)^{-3} = a^{-15}\)                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \((\ldots)^{-2}=\frac{m^{6}}{25}\)                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \((\frac{m^{3}}{5})^{-2} = \frac{m^{6}}{25}\)                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \((\ldots)^{-2}=p^{4} q^{-6}\)                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \((p^{2} q^{-3})^{-2} = p^{4} q^{-6}\)                                                                                                     
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Simplify, expressing the answers with positive indices. Evaluate powers where possible.                                               
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(3 a^{2} b^{-2}\right)^{3} \times\left(2 a^{4}\right)^{-2}\)                                                                      
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(3 a^{2} b^{-2}\right)^{3} \times\left(2 a^{4}\right)^{-2} = 27 a^{6} b^{-6} \times \frac{1}{4 a^{8}}\)\\                           
                 \(\left(3 a^{2} b^{-2}\right)^{3} \times\left(2 a^{4}\right)^{-2} = \frac{27}{4} a^{-2} b^{-6}\)\\                                         
                 \(\left(3 a^{2} b^{-2}\right)^{3} \times\left(2 a^{4}\right)^{-2} = \frac{27}{4 a^{2} b^{6}}\)                                             
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(6 a^{5} b^{-4}\right)^{-3} \times 2\left(a^{3} b^{-3}\right)^{2}\)                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\left(6 a^{5} b^{-4}\right)^{-3} \times 2\left(a^{3} b^{-3}\right)^{2} = \frac{1}{216 a^{15} b^{-12}} \times 2 a^{6} b^{-6}\)\\          
                 \(\left(6 a^{5} b^{-4}\right)^{-3} \times 2\left(a^{3} b^{-3}\right)^{2} = \frac{2}{216} a^{-9} b^{6}\)\\                                  
                 \(\left(6 a^{5} b^{-4}\right)^{-3} \times 2\left(a^{3} b^{-3}\right)^{2} = \frac{1}{108 a^{9} b^{6}}\)                                     
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{\left(2 a^{4} b^{-2}\right)^{3}}{c^{2}} \times \frac{\left(2^{2} a^{-3} b^{2}\right)^{-1}}{c}\)                                  
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\frac{\left(2 a^{4} b^{-2}\right)^{3}}{c^{2}} \times \frac{\left(2^{2} a^{-3} b^{2}\right)^{-1}}{c} = \frac{8 a^{12} b^{-6}}{c^{2}} \times \frac{c}{4 a^{3} b^{-2}}\)\\                                                                                                                               
                 \(\frac{\left(2 a^{4} b^{-2}\right)^{3}}{c^{2}} \times \frac{\left(2^{2} a^{-3} b^{2}\right)^{-1}}{c} = \frac{2 a^{9} b^{-4}}{c^{3}}\)     
             \end{solutionordottedlines}                                                                                                                    
             \part \(\mathbf{j} \frac{\left(2 a^{4}\right)^{2}}{b^{7}} \div \frac{\left(a^{2}\right)^{-3}}{2 b}\)                                           
             \begin{solutionordottedlines}[2cm]                                                                                                             
                 \(\mathbf{j} \frac{\left(2 a^{4}\right)^{2}}{b^{7}} \div \frac{\left(a^{2}\right)^{-3}}{2 b} = \mathbf{j} \frac{4 a^{8}}{b^{7}} \times     
 \frac{2 b}{a^{-6}}\)\\                                                                                                                                     
                 \(\mathbf{j} \frac{\left(2 a^{4}\right)^{2}}{b^{7}} \div \frac{\left(a^{2}\right)^{-3}}{2 b} = \mathbf{j} \frac{8 a^{14}}{b^{6}}\)         
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
     \end{questions}                                                                                                                                        
 \end{exercisebox}