 \subsection*{Section 2}                                                                                                                     
 % The Pythagorean Theorem                                                                                                                   
 \begin{questions}                                                                                                                           
     \Question[3]                                                                                                                            
     \begin{parts}                                                                                                                           
     \begin{multicols}{3}                                                                                                                    
     \part\, \drawTriangle{top left}{11}{7}{8.49}{$y$}{$7$}{$11$}{0.3}                                                                       
         \begin{solutionordottedlines}[1in]                                                                                                  
         $y = \sqrt{11^2 - 7^2} = \sqrt{121 - 49} = \sqrt{72} = 8.49$                                                                        
         \end{solutionordottedlines}                                                                                                         
     \part\, \drawTriangle{top right}{18.44}{4}{18}{$4$}{$18$}{$a$}{0.3}                                                                     
         \begin{solutionordottedlines}[1in]                                                                                                  
         $a = \sqrt{18^2 - 4^2} = \sqrt{324 - 16} = \sqrt{308} = 17.55$                                                                      
         \end{solutionordottedlines}                                                                                                         
     \part\, \drawTriangle{bottom left}{10}{7.14}{7}{$7$}{$b$}{$10$}{0.3}                                                                    
         \begin{solutionordottedlines}[1in]                                                                                                  
         $b = \sqrt{10^2 - 7^2} = \sqrt{100 - 49} = \sqrt{51} = 7.14$                                                                        
         \end{solutionordottedlines}                                                                                                         
     \end{multicols}                                                                                                                         
     \end{parts}                                                                                                                             
     \Question[3] Determine whether or not the following side-lengths form a right-angled triangle:                                          
     \begin{parts}\begin{multicols}{3}                                                                                                       
     \part $4.5,7,7.5$                                                                                                                       
         \begin{solutionordottedlines}[1in]                                                                                                  
         $7.5^2 \neq 4.5^2 + 7^2$, so it does not form a right-angled triangle.                                                              
         \end{solutionordottedlines}                                                                                                         
     \part $6,10,12$                                                                                                                         
         \begin{solutionordottedlines}[1in]                                                                                                  
         $12^2 = 6^2 + 10^2$, so it forms a right-angled triangle.                                                                           
         \end{solutionordottedlines}                                                                                                         
     \part $20,21,29$                                                                                                                        
         \begin{solutionordottedlines}[1in]                                                                                                  
         $29^2 \neq 20^2 + 21^2$, so it does not form a right-angled triangle.                                                               
         \end{solutionordottedlines}                                                                                                         
     \end{multicols}                                                                                                                         
     \end{parts}                                                                                                                             
     \Question[2] As part of a design, an artist draws a circle passing through the four corners (vertices) of a square.                     
     \begin{parts}                                                                                                                           
         \part If the square has side lengths of $4$cm, what is the radius, to the nearest millimetre, of the circle?                        
         \begin{solutionordottedlines}[1in]                                                                                                  
         Radius $r = \frac{\sqrt{4^2 + 4^2}}{2} = \frac{\sqrt{32}}{2} = \frac{4\sqrt{2}}{2} = 2\sqrt{2} \approx 2.828$cm                     
         \end{solutionordottedlines}                                                                                                         
         \part If the circle has a radius of $3$cm, what are the side lengths, to the nearest millimetre, of the square?                     
         \begin{solutionordottedlines}[1in]                                                                                                  
         Side length $s = \sqrt{2} \times r = \sqrt{2} \times 3 \approx 4.242$cm                                                             
         \end{solutionordottedlines}                                                                                                         
     \end{parts}                                                                                                                             
     \Question[2] A parent is asked to make some scarves for the local Scout troop. Two scarves can be made from one square piece of materia 
 by cutting on the diagonal. If this diagonal side length is to be $100$cm long, what must be the side length of the square piece of materia 
 to the nearest mm?                                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                  
         Side length $s = \frac{100}{\sqrt{2}} \approx 70.71$cm                                                                              
         \end{solutionordottedlines}                                                                                                         
     \Question[2] A girl planned to swim straight across a river of width $25$m. After she had swum across the river, the girl found she had 
 been swept $4$m downstream. How far did she actually swim? Calculate your answer, in metres, correct to 1 decimal place.                    
         \begin{solutionordottedlines}[1in]                                                                                                  
         Distance swum $d = \sqrt{25^2 + 4^2} \approx 25.3$m                                                                                 
         \end{solutionordottedlines}                                                                                                         
 \end{questions}                                                                                                                             
                                                                                                                                             
 \subsection*{Section 3}                                                                                                                     
 % Introduction to Surds                                                                                                                     
 \begin{questions}                                                                                                                           
     \Question[3] Use Pythagoras' Theorem to find the value of $x$. Give your answer as a \emph{surd} which has been simplified.             
     \begin{parts}                                                                                                                           
     \begin{multicols}{3}                                                                                                                    
         \part \drawTriangle{bottom left}{4.24}{3}{3}{$3$}{$x$}{$3$}{0.5}                                                                    
         \begin{solutionordottedlines}[1in]                                                                                                  
         $x = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$                                                                                      
         \end{solutionordottedlines}                                                                                                         
         \part \drawTriangle{bottom right}{2.82}{2.4}{1.73}{$\sqrt{3}$}{$\sqrt{5}$}{$x$}{0.7}                                                
         \begin{solutionordottedlines}[1in]                                                                                                  
         $x = \sqrt{\sqrt{5}^2 - \sqrt{3}^2} = \sqrt{5 - 3} = \sqrt{2}$                                                                      
         \end{solutionordottedlines}                                                                                                         
         \part \drawTriangle{bottom right}{14}{12.65}{6}{$x$}{$6$}{$14$}{0.3}                                                                
         \begin{solutionordottedlines}[1in]                                                                                                  
         $x = \sqrt{14^2 - 6^2} = \sqrt{196 - 36} = \sqrt{160} = 4\sqrt{10}$                                                                 
         \end{solutionordottedlines}                                                                                                         
     \end{multicols}                                                                                                                         
     \end{parts}                                                                                                                             
 \end{questions}                                                                                                                             
                                                                                                                                             
 \subsection*{Section 4}                                                                                                                     
 % Simplifying Surds                                                                                                                         
 \begin{questions}                                                                                                                           
     \Question[18] Simplify all of the following:                                                                                            
         \begin{parts}                                                                                                                       
         \begin{multicols}{3}                                                                                                                
         \part $(\sqrt{231})^2$                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                  
         $231$                                                                                                                               
         \end{solutionordottedlines}                                                                                                         
         \part $(3\sqrt{5})^2$                                                                                                               
         \begin{solutionordottedlines}[1in]                                                                                                  
         $9 \times 5 = 45$                                                                                                                   
         \end{solutionordottedlines}                                                                                                         
         \part $(2\sqrt{11})^2$                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                  
         $4 \times 11 = 44$                                                                                                                  
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{3}\times\sqrt{5}$                                                                                                      
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{15}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{5}\times\sqrt{6}$                                                                                                      
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{30}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{6}\div\sqrt{2}$                                                                                                        
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{3}$                                                                                                                          
         \end{solutionordottedlines}                                                                                                         
         \part $\frac{\sqrt{77}}{\sqrt{11}}$                                                                                                 
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{7}$                                                                                                                          
         \end{solutionordottedlines}                                                                                                         
         \part $2\times3\sqrt{2}$                                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                  
         $6\sqrt{2}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $6\times5\sqrt{7}$                                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                  
         $30\sqrt{7}$                                                                                                                        
         \end{solutionordottedlines}                                                                                                         
         \part $(\sqrt{2})^3$                                                                                                                
         \begin{solutionordottedlines}[1in]                                                                                                  
         $2\sqrt{2}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $(\sqrt{11})^2+(\sqrt{2})^2$                                                                                                  
         \begin{solutionordottedlines}[1in]                                                                                                  
         $11 + 2 = 13$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
         \part $(\sqrt{5})^2+(\sqrt{11})^2$                                                                                                  
         \begin{solutionordottedlines}[1in]                                                                                                  
         $5 + 11 = 16$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{45}$                                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $3\sqrt{5}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{54}$                                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $3\sqrt{6}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{126}$                                                                                                                  
         \begin{solutionordottedlines}[1in]                                                                                                  
         $3\sqrt{14}$                                                                                                                        
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{72}$                                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $6\sqrt{2}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{96}$                                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $4\sqrt{6}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
         \part $\sqrt{200}$                                                                                                                  
         \begin{solutionordottedlines}[1in]                                                                                                  
         $10\sqrt{2}$                                                                                                                        
         \end{solutionordottedlines}                                                                                                         
         \end{multicols}                                                                                                                     
         \end{parts}                                                                                                                         
         \Question[4] Express the following surds as the square root of a whole number                                                       
             \begin{parts}                                                                                                                   
             \begin{multicols}{2}                                                                                                            
                 \part $2\sqrt{3}$                                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{12}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
                 \part $2\sqrt{13}$                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{52}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
                 \part $4\sqrt{5}$                                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{80}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
                 \part $12\sqrt{10}$                                                                                                         
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\sqrt{1440}$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
             \end{multicols}                                                                                                                 
             \end{parts}                                                                                                                     
         \Question[2] Evaluate:                                                                                                              
             \begin{parts}                                                                                                                   
             \begin{multicols}{2}                                                                                                            
                 \part $(\sqrt{\frac{2}{3}})^2$                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\frac{2}{3}$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
                 \part $\sqrt{\frac{16}{25}}$                                                                                                
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\frac{4}{5}$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
             \end{multicols}                                                                                                                 
             \end{parts}                                                                                                                     
 \end{questions}                                                                                                                             
                                                                                                                                             
 \subsection*{Section 5}                                                                                                                     
 % Addition & Subtraction of Surds                                                                                                           
 \begin{questions}                                                                                                                           
     \Question[16] Simplify:                                                                                                                 
         \begin{parts}                                                                                                                       
         \begin{multicols}{2}                                                                                                                
             \part $6\sqrt{2} + 4\sqrt{2}$                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $10\sqrt{2}$                                                                                                                        
         \end{solutionordottedlines}                                                                                                         
             \part $10\sqrt{7}-5\sqrt{7}$                                                                                                    
         \begin{solutionordottedlines}[1in]                                                                                                  
         $5\sqrt{7}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
             \part $3\sqrt{11}-5\sqrt{11}$                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $-2\sqrt{11}$                                                                                                                       
         \end{solutionordottedlines}                                                                                                         
             \part $2\sqrt{2}+3\sqrt{2}+\sqrt{2}$                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                  
         $6\sqrt{2}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
             \part $10\sqrt{5}-\sqrt{5}-6\sqrt{5}$                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                  
         $3\sqrt{5}$                                                                                                                         
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{3}-2\sqrt{2}+2\sqrt{3}+\sqrt{2}$                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $3\sqrt{3}-\sqrt{2}$                                                                                                                
         \end{solutionordottedlines}                                                                                                         
             \part $5\sqrt{14}+4\sqrt{6}+\sqrt{14}+3\sqrt{6}$                                                                                
         \begin{solutionordottedlines}[1in]                                                                                                  
         $6\sqrt{14}+7\sqrt{6}$                                                                                                              
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{5}-3\sqrt{2}-4\sqrt{5}+7\sqrt{2}$                                                                                  
         \begin{solutionordottedlines}[1in]                                                                                                  
         $-3\sqrt{5}+4\sqrt{2}$                                                                                                              
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{50} + 3\sqrt{2}$                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$                                                                                                 
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{48} + 2\sqrt{3}$                                                                                                   
         \begin{solutionordottedlines}[1in]                                                                                                  
         $4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$                                                                                                 
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{8}+4\sqrt{2}+2\sqrt{18}$                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                  
         $2\sqrt{2}+4\sqrt{2}+6\sqrt{2} = 12\sqrt{2}$                                                                                        
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{8}+\sqrt{2}+\sqrt{18}$                                                                                             
         \begin{solutionordottedlines}[1in]                                                                                                  
         $2\sqrt{2}+\sqrt{2}+3\sqrt{2} = 6\sqrt{2}$                                                                                          
         \end{solutionordottedlines}                                                                                                         
             \part $4\sqrt{5}-4\sqrt{20}-\sqrt{45}$                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                  
         $4\sqrt{5}-8\sqrt{5}-3\sqrt{5} = -7\sqrt{5}$                                                                                        
         \end{solutionordottedlines}                                                                                                         
             \part \[\frac{\sqrt{7}}{2}-\frac{\sqrt{7}}{5}\]                                                                                 
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\frac{5\sqrt{7}-2\sqrt{7}}{10} = \frac{3\sqrt{7}}{10}$                                                                             
         \end{solutionordottedlines}                                                                                                         
             \part \[\frac{3\sqrt{11}}{7}+\frac{2\sqrt{11}}{21}\]                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                  
         $\frac{9\sqrt{11}+2\sqrt{11}}{21} = \frac{11\sqrt{11}}{21}$                                                                         
         \end{solutionordottedlines}                                                                                                         
             \part $\sqrt{80}-\sqrt{45} = \sqrt{x}$                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                  
         $4\sqrt{5}-3\sqrt{5} = \sqrt{5} = \sqrt{x}$, so $x = 5$                                                                             
         \end{solutionordottedlines}                                                                                                         
         \end{multicols}                                                                                                                     
         \end{parts}                                                                                                                         
 \end{questions}                                                                                                                             

 \subsection*{Section 6}                                                                                                                                             
 % Multiplication and Division of Surds                                                                                                                              
 \begin{questions}                                                                                                                                                   
     \Question[8] Simplify:                                                                                                                                          
         \begin{parts}                                                                                                                                               
         \begin{multicols}{2}                                                                                                                                        
         \part $4\sqrt{7}\times\sqrt{2}$                                                                                                                             
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $4\sqrt{7}\times\sqrt{2} = 4\sqrt{14}$                                                                                                                      
         \end{solutionordottedlines}                                                                                                                                 
         \part $6\sqrt{3}\times 4\sqrt{7}$                                                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $6\sqrt{3}\times 4\sqrt{7} = 24\sqrt{21}$                                                                                                                   
         \end{solutionordottedlines}                                                                                                                                 
         \part $12\sqrt{33}\div3\sqrt{3}$                                                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $12\sqrt{33}\div3\sqrt{3} = 4\sqrt{11}$                                                                                                                     
         \end{solutionordottedlines}                                                                                                                                 
         \part $36\sqrt{15}\div4\sqrt{3}$                                                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $36\sqrt{15}\div4\sqrt{3} = 9\sqrt{5}$                                                                                                                      
         \end{solutionordottedlines}                                                                                                                                 
         \part $7\sqrt{28}\div4\sqrt{7}$                                                                                                                             
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $7\sqrt{28}\div4\sqrt{7} = 7\sqrt{4} = 14$                                                                                                                  
         \end{solutionordottedlines}                                                                                                                                 
         \part $\sqrt{7}\times2\sqrt{7}$                                                                                                                             
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $\sqrt{7}\times2\sqrt{7} = 2\sqrt{49} = 14$                                                                                                                 
         \end{solutionordottedlines}                                                                                                                                 
         \part $7\sqrt{10}\times3\sqrt{2}$                                                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $7\sqrt{10}\times3\sqrt{2} = 21\sqrt{20} = 42\sqrt{5}$                                                                                                      
         \end{solutionordottedlines}                                                                                                                                 
         \part $\sqrt{2}(2\sqrt{2}-\sqrt{6})$                                                                                                                        
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $\sqrt{2}(2\sqrt{2}-\sqrt{6}) = 2\cdot2 - \sqrt{12} = 4 - 2\sqrt{3}$                                                                                        
         \end{solutionordottedlines}                                                                                                                                 
         \end{multicols}                                                                                                                                             
         \end{parts}                                                                                                                                                 
     \Question[12] Expand and simplify:                                                                                                                              
         \begin{parts}                                                                                                                                               
         \begin{multicols}{2}                                                                                                                                        
         \part $5\sqrt{5}(4\sqrt{2}-3)$                                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $5\sqrt{5}(4\sqrt{2}-3) = 20\sqrt{10} - 15\sqrt{5}$                                                                                                         
         \end{solutionordottedlines}                                                                                                                                 
         \part $2\sqrt{3}(3\sqrt{3}-5)$                                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $2\sqrt{3}(3\sqrt{3}-5) = 6\cdot3 - 10\sqrt{3} = 18 - 10\sqrt{3}$                                                                                           
         \end{solutionordottedlines}                                                                                                                                 
         \part $3\sqrt{7}(2-\sqrt{14})$                                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $3\sqrt{7}(2-\sqrt{14}) = 6\sqrt{7} - 3\cdot7 = 6\sqrt{7} - 21$                                                                                             
         \end{solutionordottedlines}                                                                                                                                 
         \part $\sqrt{2}(2\sqrt{2}-\sqrt{6})$                                                                                                                        
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $\sqrt{2}(2\sqrt{2}-\sqrt{6}) = 4 - 2\sqrt{3}$                                                                                                              
         \end{solutionordottedlines}                                                                                                                                 
         \part $(4\sqrt{5}+1)(3\sqrt{5}+2)$                                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(4\sqrt{5}+1)(3\sqrt{5}+2) = 12\cdot5 + 8\sqrt{5} + 3\sqrt{5} + 2 = 60 + 11\sqrt{5}$                                                                       
         \end{solutionordottedlines}                                                                                                                                 
         \part $(3\sqrt{2}+2)(3\sqrt{2}-1)$                                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(3\sqrt{2}+2)(3\sqrt{2}-1) = 9\cdot2 + 6\sqrt{2} - 3\sqrt{2} - 2 = 18 + 3\sqrt{2}$                                                                         
         \end{solutionordottedlines}                                                                                                                                 
         \part $(2\sqrt{3}-4)(3\sqrt{3}+5)$                                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(2\sqrt{3}-4)(3\sqrt{3}+5) = 6\cdot3 + 10\sqrt{3} - 12\sqrt{3} - 20 = 18 - 2\sqrt{3} - 20 = -2 - 2\sqrt{3}$                                                
         \end{solutionordottedlines}                                                                                                                                 
         \part $(7\sqrt{2}+5)^2$                                                                                                                                     
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(7\sqrt{2}+5)^2 = 49\cdot2 + 70\sqrt{2} + 25 = 98 + 70\sqrt{2} + 25 = 123 + 70\sqrt{2}$                                                                    
         \end{solutionordottedlines}                                                                                                                                 
         \part $(3\sqrt{5}+2)(\sqrt{2}+3)$                                                                                                                           
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(3\sqrt{5}+2)(\sqrt{2}+3) = 3\sqrt{10} + 9\sqrt{5} + 2\sqrt{2} + 6$                                                                                        
         \end{solutionordottedlines}                                                                                                                                 
         \part $(4+2\sqrt{3})(2\sqrt{7}-5)$                                                                                                                          
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(4+2\sqrt{3})(2\sqrt{7}-5) = 8\sqrt{7} - 20 + 4\sqrt{21} - 10\sqrt{3}$                                                                                     
         \end{solutionordottedlines}                                                                                                                                 
         \part $(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})$                                                                                                              
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5}) = 7 - 5 = 2$                                                                                                        
         \end{solutionordottedlines}                                                                                                                                 
         \part $(4\sqrt{7}-\sqrt{5})(2\sqrt{5}+\sqrt{7})$                                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $(4\sqrt{7}-\sqrt{5})(2\sqrt{5}+\sqrt{7}) = 8\sqrt{35} + 4\cdot7 - 2\cdot5 - \sqrt{35} = 28 - 10 + 7\sqrt{35}$                                              
         \end{solutionordottedlines}                                                                                                                                 
         \end{multicols}                                                                                                                                             
         \end{parts}                                                                                                                                                 
         \question \textbf{Challenge:}                                                                                                                               
         \Question[2] If $x = \sqrt{2} - 1$ and $y = \sqrt{3} + 1$, find:                                                                                            
         \begin{solutionordottedlines}[1in]                                                                                                                          
         $x + y = (\sqrt{2} - 1) + (\sqrt{3} + 1) = \sqrt{2} + \sqrt{3}$                                                                                             
         \end{solutionordottedlines}                                                                                                                                 
         \Question[2] \[\frac{6\sqrt{10}}{x+1}\]                                                                                                                     
         \begin{solutionordottedlines}[1.5in]                                                                                                                        
         \[\frac{6\sqrt{10}}{\sqrt{2}} = 6\sqrt{5}\]                                                                                                                 
         \end{solutionordottedlines}                                                                                                                                 
         \Question[2] \[x + \frac{1}{x}\]                                                                                                                            
         \begin{solutionordottedlines}[1.5in]                                                                                                                        
         \[\sqrt{2} - 1 + \frac{1}{\sqrt{2} - 1} = \sqrt{2} - 1 + \frac{\sqrt{2} + 1}{1} = 2\sqrt{2}\]                                                               
         \end{solutionordottedlines}                                                                                                                                 
 \end{questions}                                                                                                                                                     
                                                                                                                                                                     
