                                                                                                                                      
 \begin{examplebox}                                                                                                                   
     \subsection{Examples:}                                                                                                           
     \begin{questions}                                                                                                                
         \Question[2] Express the following with a \emph{rational} denominator                                                        
         \begin{doublespace}                                                                                                          
         \begin{parts}                                                                                                                
             \part \(\frac{4}{\sqrt{2}}=\)                                                                                            
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}\)                            
                 \end{solutionordottedlines}                                                                                          
             \part \(\frac{9}{4 \sqrt{3}}=\)                                                                                          
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{9}{4\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{9\sqrt{3}}{12} = \frac{3\sqrt{3}}{4}\)                
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{2}{\sqrt{3}}+\frac{\sqrt{3}}{2}=\]                                                                         
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} + \frac{\sqrt{3}}{2} \times \frac{2}{2} = \frac{2\sqrt{3}}{3}  
 \frac{2\sqrt{3}}{4} = \frac{4\sqrt{3} + 3\sqrt{3}}{6} = \frac{7\sqrt{3}}{6}\)                                                        
                 \end{solutionordottedlines}                                                                                          
         \end{parts}                                                                                                                  
         \end{doublespace}                                                                                                            
     \end{questions}                                                                                                                  
 \end{examplebox}                                                                                                                     
                                                                                                                                      
 \begin{exercisebox}                                                                                                                  
     \subsection{Exercises:}                                                                                                          
     \begin{questions}                                                                                                                
         \Question[6]                                                                                                                 
         \begin{parts}\begin{multicols}{2}                                                                                            
             \begin{doublespace}                                                                                                      
             \part \[\frac{\sqrt{3}}{\sqrt{7}}=\]                                                                                     
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{\sqrt{3}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{21}}{7}\)                                 
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{2 \sqrt{7}}{\sqrt{3}}=\]                                                                                   
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{2 \sqrt{7}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{21}}{3}\)                              
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{\sqrt{2}}{3 \sqrt{10}}=\]                                                                                  
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{\sqrt{2}}{3\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{20}}{30} = \frac{\sqrt{5}}{15}\)      
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{\sqrt{15}}{3 \sqrt{5}}=\]                                                                                  
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{\sqrt{15}}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{75}}{15} = \frac{5}{3}\)                
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{2}{\sqrt{3}}+\frac{3}{2 \sqrt{3}}=\]                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} + \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} =       
 \frac{2\sqrt{3}}{3} + \frac{3\sqrt{3}}{6} = \frac{4\sqrt{3} + 3\sqrt{3}}{6} = \frac{7\sqrt{3}}{6}\)                                  
                 \end{solutionordottedlines}                                                                                          
             \part \[\frac{5 \sqrt{2}}{3}-\frac{1}{\sqrt{3}}=\]                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 \(\frac{5 \sqrt{2}}{3} - \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5 \sqrt{2}}{3} -                
 \frac{\sqrt{3}}{3} = \frac{5\sqrt{2} - \sqrt{3}}{3}\)                                                                                
                 \end{solutionordottedlines}                                                                                          
             \end{doublespace}                                                                                                        
         \end{multicols}\end{parts}                                                                                                   
         \Question[4] Find the value of \(x\). Express your answer with a rational denominator.                                       
             \begin{parts}                                                                                                            
                 \begin{multicols}{2}                                                                                                 
                     \part \includegraphics[width=0.4\linewidth]{img/4a}                                                              
                         \begin{solutionordottedlines}[4cm]                                                                           
                         % Since the image is not available, I cannot provide a solution.                                             
                         \end{solutionordottedlines}                                                                                  
                     \part \includegraphics[width=0.4\linewidth]{img/4b}                                                              
                         \begin{solutionordottedlines}[4cm]                                                                           
                         % Since the image is not available, I cannot provide a solution.                                             
                         \end{solutionordottedlines}                                                                                  
                 \end{multicols}                                                                                                      
             \end{parts}                                                                                                              
     \end{questions}                                                                                                                  
 \end{exercisebox}                                                                                                                    
                                                                                                                                      