                                                                                                                                      
 As promised, here lie the \emph{binomial denominators}. There are a couple of prerequisites to understanding this mathematics, the   
 first is that you must totally grasp $(a+b)(a-b) = a^2 - b^2$.                                                                       
                                                                                                                                      
 The significance of this result is that if we make $a$ and $b$ surds like so: $(\sqrt{a}+\sqrt{b})$, then multiplying by             
 $(\sqrt{a}-\sqrt{b})$ yields a fantastic result, namely:                                                                             
                                                                                                                                      
 \vspace{0.5cm}                                                                                                                       
 \begin{equation}                                                                                                                     
     (\source{\sqrt{a}}+\source{\sqrt{b}})(\target{\sqrt{a}}-\target{\sqrt{b}})= (\sqrt{a})^2 - (\sqrt{b})^2\mbox{\drawarrows}\\      
     = a - b                                                                                                                          
 \end{equation}                                                                                                                       
 \vspace{0.5cm}                                                                                                                       
                                                                                                                                      
 This act of starting with $\sqrt{a} + \sqrt{b}$, and multiplying it with the sign flipped: $\sqrt{a}-\sqrt{b}$ is called             
 \emph{multiplying by the conjugate}. Let's practise this method:                                                                     
                                                                                                                                      
 \begin{examplebox}                                                                                                                   
     \subsection{Examples:}                                                                                                           
     \begin{questions}                                                                                                                
         \Question[1] \[\frac{2 \sqrt{5}}{2 \sqrt{5}-2}=\]                                                                            
             \begin{solutionordottedlines}[3cm]                                                                                       
                 Multiply by the conjugate: $\frac{2 \sqrt{5}}{2 \sqrt{5}-2} \cdot \frac{2 \sqrt{5}+2}{2 \sqrt{5}+2} = \frac{4 \cdot  
 + 4 \sqrt{5}}{20 - 4} = \frac{20 + 4 \sqrt{5}}{16} = \frac{5}{4} + \sqrt{5}$                                                         
             \end{solutionordottedlines}                                                                                              
         \Question[1] \[\frac{\sqrt{3}+\sqrt{2}}{3 \sqrt{2}+2 \sqrt{3}}\]                                                             
             \begin{solutionordottedlines}[3cm]                                                                                       
                 Multiply by the conjugate: $\frac{\sqrt{3}+\sqrt{2}}{3 \sqrt{2}+2 \sqrt{3}} \cdot \frac{3 \sqrt{2}-2 \sqrt{3}}{3     
 \sqrt{2}-2 \sqrt{3}} = \frac{3 \cdot 2 - 2 \cdot 3}{6 - 4 \sqrt{6} + 4 \sqrt{6} - 12} = \frac{6 - 6}{-6} = 0$                        
             \end{solutionordottedlines}                                                                                              
     \end{questions}                                                                                                                  
 \end{examplebox}                                                                                                                     
                                                                                                                                      
                                                                                                                                      
\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[1] \[\frac{2 \sqrt{5}}{2 \sqrt{5}-2}=\]
            \begin{solutionordottedlines}[3cm]
            \end{solutionordottedlines}
        \Question[1] \[\frac{\sqrt{3}+\sqrt{2}}{3 \sqrt{2}+2 \sqrt{3}}\]
            \begin{solutionordottedlines}[3cm]
            \end{solutionordottedlines}
    \end{questions}
\end{examplebox}
                                                                                                                                      
