This is an extension of the things you learned in \emph{Topic 1: Algebra}. We will be using 3 main results and then replacing the $a$'s and $b$'s with \emph{surds}.

\begin{align}
& (a+b)^{2}=a^{2}+2 a b+b^{2} \\
& (a-b)^{2}=a^{2}-2 a b+b^{2} \\
& (a-b)(a+b)=a^{2}-b^{2}
\end{align}

 \begin{examplebox}                                                                                                                   
     \subsection{Examples:}                                                                                                           
     \begin{questions}                                                                                                                
         \Question[5]                                                                                                                 
         \begin{doublespace}                                                                                                          
         \begin{parts}                                                                                                                
         \part \[(\sqrt{5}+\sqrt{3})^{2}=\]                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(\sqrt{5}+\sqrt{3})^{2} = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$    
             \end{solutionordottedlines}                                                                                              
         \part \[(3 \sqrt{2}-2 \sqrt{3})^{2}=\]                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(3 \sqrt{2}-2 \sqrt{3})^{2} = (3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 9\cdot2 - 2\cdot6\sqrt{6} +     
 4\cdot3 = 18 - 12\sqrt{6} + 12$                                                                                                      
             \end{solutionordottedlines}                                                                                              
         \part \[(2 \sqrt{3}+4 \sqrt{6})^{2}=\]                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(2 \sqrt{3}+4 \sqrt{6})^{2} = (2\sqrt{3})^2 + 2(2\sqrt{3})(4\sqrt{6}) + (4\sqrt{6})^2 = 4\cdot3 + 2\cdot8\sqrt{18} +    
 16\cdot6 = 12 + 16\sqrt{18} + 96$                                                                                                    
             \end{solutionordottedlines}                                                                                              
         \part \[(\sqrt{7}-5)(\sqrt{7}+5)=\]                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(\sqrt{7}-5)(\sqrt{7}+5) = (\sqrt{7})^2 - (5)^2 = 7 - 25 = -18$                                                         
             \end{solutionordottedlines}                                                                                              
         \part \[(5 \sqrt{6}-2 \sqrt{5})(5 \sqrt{6}+2 \sqrt{5})=\]                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(5 \sqrt{6}-2 \sqrt{5})(5 \sqrt{6}+2 \sqrt{5}) = (5\sqrt{6})^2 - (2\sqrt{5})^2 = 25\cdot6 - 4\cdot5 = 150 - 20 = 130$   
             \end{solutionordottedlines}                                                                                              
         \end{parts}                                                                                                                  
         \end{doublespace}                                                                                                            
     \end{questions}                                                                                                                  
 \end{examplebox}                                                                                                                     

 \begin{exercisebox}                                                                                                                  
     \subsection{Exercises:}                                                                                                          
     \begin{questions}                                                                                                                
         \Question[12]                                                                                                                
         \begin{parts}\begin{multicols}{2}                                                                                            
             \part \((5+\sqrt{3})^{2}\)                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(5+\sqrt{3})^{2} = 5^2 + 2(5)(\sqrt{3}) + (\sqrt{3})^2 = 25 + 10\sqrt{3} + 3 = 28 + 10\sqrt{3}$                         
             \end{solutionordottedlines}                                                                                              
             \part \((\sqrt{2}+6)^{2}\)                                                                                               
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(\sqrt{2}+6)^{2} = (\sqrt{2})^2 + 2(\sqrt{2})(6) + 6^2 = 2 + 12\sqrt{2} + 36 = 38 + 12\sqrt{2}$                     
                 \end{solutionordottedlines}                                                                                          
             \part \((5 \sqrt{6}+2 \sqrt{3})^{2}\)                                                                                    
                 \begin{solutionordottedlines}[2cm]                                                                                   
                     $(5 \sqrt{6}+2 \sqrt{3})^{2} = (5\sqrt{6})^2 + 2(5\sqrt{6})(2\sqrt{3}) + (2\sqrt{3})^2 = 25\cdot6 + 2\cdot10\sqrt{18} + 4\cdot3 = 150 + 20\sqrt{18} + 12$
                 \end{solutionordottedlines}                                                                                          
             \part \((\sqrt{21}+\sqrt{3})^{2}\)                                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(\sqrt{21}+\sqrt{3})^{2} = (\sqrt{21})^2 + 2(\sqrt{21})(\sqrt{3}) + (\sqrt{3})^2 = 21 + 2\sqrt{63} + 3 = 24 + 2\sqrt{63}$
                 \end{solutionordottedlines}                                                                                          
             \part \((4+2 \sqrt{5})^{2}\)                                                                                             
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(4+2 \sqrt{5})^{2} = 4^2 + 2(4)(2\sqrt{5}) + (2\sqrt{5})^2 = 16 + 16\sqrt{5} + 4\cdot5 = 16 + 16\sqrt{5} + 20 = 36  16\sqrt{5}$ 
                 \end{solutionordottedlines}                                                                                          
             \part \((\sqrt{7}-2)^{2}\)                                                                                               
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(\sqrt{7}-2)^{2} = (\sqrt{7})^2 - 2(\sqrt{7})(2) + 2^2 = 7 - 4\sqrt{7} + 4 = 11 - 4\sqrt{7}$                        
                 \end{solutionordottedlines}                                                                                          
             \part \((4-\sqrt{3})^{2}\)                                                                                               
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(4-\sqrt{3})^{2} = 4^2 - 2(4)(\sqrt{3}) + (\sqrt{3})^2 = 16 - 8\sqrt{3} + 3 = 19 - 8\sqrt{3}$                       
                 \end{solutionordottedlines}                                                                                          
             \part \((2 \sqrt{35}+\sqrt{5})^{2}\)                                                                                     
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(2 \sqrt{35}+\sqrt{5})^{2} = (2\sqrt{35})^2 + 2(2\sqrt{35})(\sqrt{5}) + (\sqrt{5})^2 = 4\cdot35 + 2\cdot2\sqrt{175} 5 = 140 + 4\sqrt{175} + 5$ 
                 \end{solutionordottedlines}                                                                                          
             \part \((\sqrt{70}-3 \sqrt{10})^{2}\)                                                                                    
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(\sqrt{70}-3 \sqrt{10})^{2} = (\sqrt{70})^2 - 2(\sqrt{70})(3\sqrt{10}) + (3\sqrt{10})^2 = 70 - 2\cdot3\sqrt{700} + 9\cdot10 = 70 - 6\sqrt{700} + 90$
                 \end{solutionordottedlines}                                                                                          
             \part \((3-\sqrt{5})(3+\sqrt{5})\)                                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(3-\sqrt{5})(3+\sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4$                                                          
                 \end{solutionordottedlines}                                                                                          
             \part \((2 \sqrt{5}-1)^{2}\)                                                                                             
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(2 \sqrt{5}-1)^{2} = (2\sqrt{5})^2 - 2(2\sqrt{5})(1) + 1^2 = 4\cdot5 - 2\cdot2\sqrt{5} + 1 = 20 - 4\sqrt{5} + 1 = 2 - 4\sqrt{5}$
                 \end{solutionordottedlines}                                                                                          
             \part \((\sqrt{6}-1)(\sqrt{6}+1)\)                                                                                       
                 \begin{solutionordottedlines}[2cm]                                                                                   
                 $(\sqrt{6}-1)(\sqrt{6}+1) = (\sqrt{6})^2 - 1^2 = 6 - 1 = 5$                                                          
                 \end{solutionordottedlines}                                                                                          
             %\item \((2 \sqrt{3}-5 \sqrt{6})(2 \sqrt{3}+5 \sqrt{6})\)                                                                
             %    \mdots{2}                                                                                                           
             %\item \((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})\)                                                                        
             %    \mdots{2}                                                                                                           
         \end{multicols}\end{parts}                                                                                                   
         \end{questions}                                                                                                              
 \end{exercisebox}                                                                                                                    
