You will want to pay attention to this part, because we will come up against a more involved version of \emph{rationalising the denominator} in about 20 minutes. We will begin to see them not just as $\frac{1}{\sqrt{2}}$ as we see them in this section, but rather as \emph{binomials}: $\frac{1}{\sqrt{3}-\sqrt{2}}$.

We explained the \underline{motivations} behind rationalising the denominator in the introduction but a quick revision will not hurt: \\

\begin{center}
\fbox{
    \begin{minipage}{0.3\textwidth}
\begin{enumerate}[(1)]
    \item Historical Convention 
    \item Simplicity 
    \item Aesthetics 
    \item Clarity
\end{enumerate}
    \end{minipage}
}
\end{center}

Now having said this, what is the mathematical method to rationalise a denominator? Simple, just multiply the denominator with whatever you need to such that the surd cancels out:

\[
\begin{aligned}
    \frac{2}{\sqrt{2}} &= \frac{2 \textcolor{red}{\times \sqrt{2}}}{\sqrt{2} \textcolor{red}{\times \sqrt{2}}}\\
    &= \frac{\cancel{2}\sqrt{2}}{\cancel{2}}\\
    &= \sqrt{2}
\end{aligned}
\]

\textbf{Recall that:} you may multiply or divide a fraction by any number as long as you \emph{do the top what you do to the bottom}. 

 \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{theorembox}
    \subsection*{Formulae:}
\[
\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}, \text { where } a \text { is positive, }
\]

\end{theorembox}

 \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}