For the moment we are finished with \emph{geometric} thinking and are now going to begin playing with numbers and doing arithmetic on terms such as $\sqrt{121}$ and $\sqrt{48}$. You should be able to simplify the former $\sqrt{121} = \fillin[11]$, but the latter is a little more tricky.

Watch this: \(\sqrt{48} = \sqrt{16\times3} = 4\sqrt{3}\). This surd is now considered simplified.


 \begin{examplebox}                                                                                                                          
     \section{Examples}                                                                                                                     
     Try to simplify the following:                                                                                                          
     \begin{multicols}{2}                                                                                                                    
         \begin{questions}                                                                                                                   
             \Question[1] $\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}$                                                                              
             \Question[1] $\sqrt{108}=\sqrt{36\times3}=6\sqrt{3}$                                                                            
             \Question[1] $\sqrt{45}=\sqrt{9\times5}=3\sqrt{5}$                                                                              
             \Question[1] $\sqrt{32}=\sqrt{16\times2}=4\sqrt{2}$                                                                             
         \end{questions}                                                                                                                     
     \end{multicols}                                                                                                                         
 \end{examplebox}                                                                                                                            

 \begin{exercisebox}                                                                                                                         
     \section{Exercises}                                                                                                                    
     Now try going backwards:                                                                                                                
     \begin{multicols}{2}                                                                                                                    
         \begin{questions}                                                                                                                   
             \Question[1] $3\sqrt{7}=\sqrt{9\times7}=\sqrt{63}$                                                                              
             \Question[1] $5\sqrt{3}=\sqrt{25\times3}=\sqrt{75}$                                                                             
         \end{questions}                                                                                                                     
     \end{multicols}                                                                                                                         
 \end{exercisebox}                                                                                                                           

You will be doing much more practise with these new mathematical objects in the homework, so do not worry if it feels uncomfortable right now!
