 \begin{problembox}                                                                                                             
     \subsection{Formulae}                                                                                                      
     \begin{questions}                                                                                                          
         \Question[3] Temperatures can be measured in either degrees Fahrenheit or degrees Celsius. To convert from one scale t 
 the other, the following formula is used: \(F=\frac{9}{5} C+32\).                                                              
         \begin{parts}                                                                                                          
             \part Rearrange the formula to make \(C\) the subject.                                                             
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(C = \frac{5}{9}(F - 32)\)                                                                                    
             \end{solutionordottedlines}                                                                                        
             \part On a particular day in Melbourne, the temperature was \(28^{\circ} \mathrm{C}\). What is this temperature    
 measured in Fahrenheit?                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(F = \frac{9}{5} \times 28 + 32 = 82.4^{\circ} \mathrm{F}\)                                                   
             \end{solutionordottedlines}                                                                                        
             \part In Boston, USA, the minimum overnight temperature was \(4^{\circ} \mathrm{F}\). What is this temperature     
 measured in Celsius?                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(C = \frac{5}{9}(4 - 32) = -15.56^{\circ} \mathrm{C}\)                                                        
             \end{solutionordottedlines}                                                                                        
             \part What number represents the same temperature in \({ }^{\circ} \mathrm{C}\) and \({ }^{\circ} \mathrm{F}\) ?   
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(C = F\) \\                                                                                                   
                 \(\frac{5}{9}(F - 32) = F\) \\                                                                                 
                 \(5F - 160 = 9F\) \\                                                                                           
                 \(4F = 160\) \\                                                                                                
                 \(F = 40^{\circ}\)                                                                                             
             \end{solutionordottedlines}                                                                                        
             \part An approximate conversion formula, used frequently when converting oven temperatures, is \(F=2 C+30\). Use   
 this to convert these temperatures:                                                                                            
             \begin{subparts}                                                                                                   
                 \subpart an oven temperature of \(180^{\circ} \mathrm{C}\)                                                     
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(F = 2 \times 180 + 30 = 390^{\circ} \mathrm{F}\)                                                             
             \end{solutionordottedlines}                                                                                        
                 \subpart an oven temperature of \(530^{\circ} \mathrm{F}\)                                                     
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(C = \frac{530 - 30}{2} = 250^{\circ} \mathrm{C}\)                                                            
             \end{solutionordottedlines}                                                                                        
             \end{subparts}                                                                                                     
         \end{parts}                                                                                                            
         \Question[2] Gareth the gardener has a large rectangular vegetable patch and he wishes to put in a path around it usin 
 concrete pavers that measure \(50 \mathrm{~cm} \times 50 \mathrm{~cm}\). The path is to be 1 paver wide. Let \(n\) be the numb 
 of pavers required. If the vegetable patch measures \(x\) metres by \(y\) metres, find a formula for \(n\) in terms of \(x\) a 
 \(y\).                                                                                                                         
         \begin{solutionordottedlines}[2cm]                                                                                     
             \(n = 2(x + y) \times 2\) because each side needs \(x \times 2\) and \(y \times 2\) pavers, and there are two      
 lengths and two widths.                                                                                                        
         \end{solutionordottedlines}                                                                                            
         \Question[3] If \(s=\frac{n}{2}(2 a+(n-1) d)\) :                                                                       
         \begin{parts}                                                                                                          
             \part find the value of \(s\) when \(n=10, a=6\) and \(d=3\)                                                       
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(s = \frac{10}{2}(2 \times 6 + (10 - 1) \times 3) = 5(12 + 27) = 5 \times 39 = 195\)                          
             \end{solutionordottedlines}                                                                                        
             \part find the value of \(a\) when \(s=350, n=20\) and \(d=4\)                                                     
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(350 = \frac{20}{2}(2a + (20 - 1) \times 4)\) \\                                                              
                 \(350 = 10(2a + 76)\) \\                                                                                       
                 \(35 = 2a + 76\) \\                                                                                            
                 \(a = \frac{35 - 76}{2} = -20.5\)                                                                              
             \end{solutionordottedlines}                                                                                        
             \part find the value of \(d\) when \(s=460, n=10\) and \(a=10\)                                                    
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(460 = \frac{10}{2}(2 \times 10 + (10 - 1)d)\) \\                                                             
                 \(460 = 5(20 + 9d)\) \\                                                                                        
                 \(92 = 20 + 9d\) \\                                                                                            
                 \(d = \frac{92 - 20}{9} = 8\)                                                                                  
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] The formula for the geometric mean \(m\) of two positive numbers \(a\) and \(b\) is \(m=\sqrt{a b}\).     
         \begin{parts}                                                                                                          
             \part Find \(m\) if \(a=16\) and \(b=25\).                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(m = \sqrt{16 \times 25} = \sqrt{400} = 20\)                                                                  
             \end{solutionordottedlines}                                                                                        
             \part Find \(a\) if \(m=7\) and \(b=16\).                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(7 = \sqrt{a \times 16}\) \\                                                                                  
                 \(49 = a \times 16\) \\                                                                                        
                 \(a = \frac{49}{16} = 3.0625\)                                                                                 
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] If \(x=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\) :                                                              
         \begin{parts}                                                                                                          
             \part find \(x\) if \(b=4, a=1\) and \(c=-24\)                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(x = \frac{-4 + \sqrt{4^2 - 4 \times 1 \times (-24)}}{2 \times 1}\) \\                                        
                 \(x = \frac{-4 + \sqrt{16 + 96}}{2}\) \\                                                                       
                 \(x = \frac{-4 + \sqrt{112}}{2}\) \\                                                                           
                 \(x = \frac{-4 + 10.58}{2}\) \\                                                                                
                 \(x = 3.29\)                                                                                                   
             \end{solutionordottedlines}                                                                                        
             \part find \(c\) if \(a=1, x=6\) and \(b=2\)                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(6 = \frac{-2 + \sqrt{2^2 - 4 \times 1 \times c}}{2 \times 1}\) \\                                            
                 \(12 = -2 + \sqrt{4 - 4c}\) \\                                                                                 
                 \(14 = \sqrt{4 - 4c}\) \\                                                                                      
                 \(196 = 4 - 4c\) \\                                                                                            
                 \(c = \frac{4 - 196}{4} = -48\)                                                                                
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] A pillar is in the shape of a cylinder with a hemispherical top. If \(r\) metres is the radius of the bas 
 and \(h\) metres is the total height, the volume \(V\) cubic metres is given by the formula \(V=\frac{1}{3} \pi r^{2}(3 h-r)\) 
         \begin{parts}                                                                                                          
             \part Rearrange the formula to make \(h\) the subject.                                                             
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(V = \frac{1}{3} \pi r^2 (3h - r)\) \\                                                                        
                 \(3V = \pi r^2 (3h - r)\) \\                                                                                   
                 \(3V = 3\pi r^2 h - \pi r^3\) \\                                                                               
                 \(3V + \pi r^3 = 3\pi r^2 h\) \\                                                                               
                 \(h = \frac{3V + \pi r^3}{3\pi r^2}\)                                                                          
             \end{solutionordottedlines}                                                                                        
             \part Find the height of the pillar, correct to the nearest centimetre, if the radius of the pillar is \(0.5       
 \mathrm{~m}\) and the volume is \(10 \mathrm{~m}^{3}\).                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(h = \frac{3 \times 10 + \pi \times 0.5^3}{3\pi \times 0.5^2}\) \\                                            
                 \(h = \frac{30 + \frac{1}{8}\pi}{\frac{3}{4}\pi}\) \\                                                          
                 \(h = \frac{30 + 0.3927}{2.3562}\) \\                                                                          
                 \(h = \frac{30.3927}{2.3562}\) \\                                                                              
                 \(h = 12.90 \mathrm{~m}\) \\                                                                                   
                 \(h \approx 1290 \mathrm{~cm}\)                                                                                
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[3] Rearrange each of these formulas to make the pronumeral in brackets the subject. (All of the pronumerals  
 represent positive numbers.)                                                                                                   
         \begin{parts}                                                                                                          
             \part \(A=\ell \times w\hfill{}(\ell)\)                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(\ell = \frac{A}{w}\)                                                                                         
             \end{solutionordottedlines}                                                                                        
             \part \(V=\pi r^{2} h\hfill{}(r)\)                                                                                 
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(r = \sqrt{\frac{V}{\pi h}}\)                                                                                 
             \end{solutionordottedlines}                                                                                        
             \part \(\frac{1}{x}+\frac{1}{y}=\frac{2}{z}\hfill{}(z)\)                                                           
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(\frac{1}{x}+\frac{1}{y}=\frac{2}{z}\) \\                                                                     
                 \(\frac{y+x}{xy}=\frac{2}{z}\) \\                                                                              
                 \(z = \frac{2xy}{x+y}\)                                                                                        
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] If a stone is dropped off a cliff, the number of metres it has fallen after a certain number of seconds i 
 found by multiplying the square of the number of seconds by 4.9.                                                               
         \begin{parts}                                                                                                          
             \part Find the formula for the distance \(d\) metres fallen by the stone in \(t\) seconds.                         
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(d = 4.9t^2\)                                                                                                 
             \end{solutionordottedlines}                                                                                        
             \part Find the distance fallen in 1.5 seconds.                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(d = 4.9 \times 1.5^2 = 4.9 \times 2.25 = 11.025 \mathrm{~m}\)                                                
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[3] If \(t=\sqrt{\frac{M}{M-m}}\) :                                                                           
         \begin{parts}                                                                                                          
             \part express the formula with \(m\) as the subject                                                                
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(t^2 = \frac{M}{M-m}\) \\                                                                                     
                 \(t^2(M-m) = M\) \\                                                                                            
                 \(t^2M - t^2m = M\) \\                                                                                         
                 \(t^2m = t^2M - M\) \\                                                                                         
                 \(m = \frac{t^2M - M}{t^2}\)                                                                                   
             \end{solutionordottedlines}                                                                                        
             \part express the formula with \(M\) as the subject                                                                
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(t^2 = \frac{M}{M-m}\) \\                                                                                     
                 \(t^2(M-m) = M\) \\                                                                                            
                 \(M - t^2m = M/t^2\) \\                                                                                        
                 \(M(1 - 1/t^2) = t^2m\) \\                                                                                     
                 \(M = \frac{t^2m}{1 - 1/t^2}\)                                                                                 
             \end{solutionordottedlines}                                                                                        
             \part find the value of \(M\) if \(m=3\) and \(t=\sqrt{2}\).                                                       
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(M = \frac{(\sqrt{2})^2 \times 3}{1 - 1/(\sqrt{2})^2}\) \\                                                    
                 \(M = \frac{2 \times 3}{1 - 1/2}\) \\                                                                          
                 \(M = \frac{6}{1/2}\) \\                                                                                       
                 \(M = 12\)                                                                                                     
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] The total surface area \(S \mathrm{~cm}^{2}\) of a cylinder is given in terms of its radius \(r           
 \mathrm{~cm}\) and height \(h \mathrm{~cm}\) by the formula \(S=2 \pi r(r+h)\).                                                
         \begin{parts}                                                                                                          
             \part Express this formula with \(h\) as the subject.                                                              
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(S = 2 \pi r^2 + 2 \pi r h\) \\                                                                               
                 \(h = \frac{S - 2 \pi r^2}{2 \pi r}\)                                                                          
             \end{solutionordottedlines}                                                                                        
             \part What is the height of such a cylinder if the radius is \(7 \mathrm{~cm}\) and the total surface area is \(50 
 \mathrm{~cm}^{2}\) ? Calculate your answer in centimetres, correct to 2 decimal places.                                        
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(h = \frac{500 - 2 \pi \times 7^2}{2 \pi \times 7}\) \\                                                       
                 \(h = \frac{500 - 2 \pi \times 49}{2 \pi \times 7}\) \\                                                        
                 \(h = \frac{500 - 307.88}{43.98}\) \\                                                                          
                 \(h = \frac{192.12}{43.98}\) \\                                                                                
                 \(h = 4.37 \mathrm{~cm}\)                                                                                      
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] The sum \(S\) of the squares of the first \(n\) whole numbers is given by the formula \(S=\frac{n(n+1)(2  
 n+1)}{6}\). Find the sum of the squares of:                                                                                    
         \begin{parts}                                                                                                          
             \part the first 20 whole numbers                                                                                   
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(S = \frac{20(20+1)(2 \times 20+1)}{6}\) \\                                                                   
                 \(S = \frac{20 \times 21 \times 41}{6}\) \\                                                                    
                 \(S = \frac{17220}{6}\) \\                                                                                     
                 \(S = 2870\)                                                                                                   
             \end{solutionordottedlines}                                                                                        
             \part all the numbers from 5 to 21 inclusive                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                 
                 \(S_{21} = \frac{21(21+1)(2 \times 21+1)}{6}\) \\                                                              
                 \(S_{21} = \frac{21 \times 22 \times 43}{6}\) \\                                                               
                 \(S_{21} = 3311\) \\                                                                                           
                 \(S_{4} = \frac{4(4+1)(2 \times 4+1)}{6}\) \\                                                                  
                 \(S_{4} = \frac{4 \times 5 \times 9}{6}\) \\                                                                   
                 \(S_{4} = 30\) \\                                                                                              
                 \(S = S_{21} - S_{4} = 3311 - 30 = 3281\)                                                                      
             \end{solutionordottedlines}                                                                                        
         \end{parts}                                                                                                            
         \Question[2] For the formula \(D=\sqrt{\frac{f+x}{f-x}}\), make \(x\) the subject.                                     
         \begin{solutionordottedlines}[2cm]                                                                                     
             \(D^2 = \frac{f+x}{f-x}\) \\                                                                                       
             \(D^2(f-x) = f+x\) \\                                                                                              
             \(D^2f - D^2x = f+x\) \\                                                                                           
             \(D^2x + x = D^2f - f\) \\                                                                                         
             \(x(D^2 + 1) = D^2f - f\) \\                                                                                       
             \(x = \frac{D^2f - f}{D^2 + 1}\)                                                                                   
         \end{solutionordottedlines}                                                                                            
     \Question[2] Cans in a supermarket are displayed in a triangular stack with one can at the top, two cans in the second row from the top, three 
 cans in the third row from the top, and so on. What is the number of cans in the display if the number of rows is:                                 
         \begin{parts}                                                                                                                              
             \part \(n\)                                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 The number of cans is given by the \(n\)th triangular number, which is \(\frac{n(n+1)}{2}\).                                       
             \end{solutionordottedlines}                                                                                                            
             \part 35                                                                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 For \(n = 35\), the number of cans is \(\frac{35(35+1)}{2} = \frac{35 \times 36}{2} = 630\).                                       
             \end{solutionordottedlines}                                                                                                            
         \end{parts}                                                                                                                                
     \Question[2] Rearrange this formula to make the pronumeral in brackets the subject. \[v^{2}=u^{2}+2 a s\hfill{}(u)\]                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             To make \(u\) the subject, we rearrange the formula: \(u = \sqrt{v^{2} - 2 a s}\).                                                     
         \end{solutionordottedlines}                                                                                                                
     \Question[2] For the formula \(I=\frac{180 n-360}{n}\) :                                                                                       
         \begin{parts}                                                                                                                              
             \part find \(I\) when \(n=6\)                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 For \(n = 6\), \(I = \frac{180 \times 6 - 360}{6} = \frac{1080 - 360}{6} = \frac{720}{6} = 120\).                                  
             \end{solutionordottedlines}                                                                                                            
             \part make \(n\) the subject of the formula and find \(n\) when \(I=108\)                                                              
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 To make \(n\) the subject, we rearrange the formula: \(n = \frac{180}{I + 2}\). For \(I = 108\), \(n = \frac{180}{108 + 2} =       
 \frac{180}{110} = \frac{18}{11}\).                                                                                                                 
             \end{solutionordottedlines}                                                                                                            
         \end{parts}                                                                                                                                
     \Question[4] Find the formula connecting \(x\) and \(y\) for each of these statements making \(y\) the subject.                                
         \begin{parts}                                                                                                                              
             \part \(y\) is four more than twice the square of \(x\).                                                                               
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 \(y = 2x^{2} + 4\).                                                                                                                
             \end{solutionordottedlines}                                                                                                            
             \part \(x\) and \(y\) are complementary angles.                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 \(y = 90 - x\).                                                                                                                    
             \end{solutionordottedlines}                                                                                                            
             \part A car travelled \(x \mathrm{~km}\) in \(y\) hours at a speed of \(100 \mathrm{~km} / \mathrm{h}\).                               
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 \(y = \frac{x}{100}\).                                                                                                             
             \end{solutionordottedlines}                                                                                                            
             \part A car travelled \(100 \mathrm{~km}\) in \(y\) hours at a speed of \(x \mathrm{~km} / \mathrm{h}\).                               
             \begin{solutionordottedlines}[2cm]                                                                                                     
                 \(y = \frac{100}{x}\).                                                                                                             
             \end{solutionordottedlines}                                                                                                            
         \end{parts}                                                                                                                                
     \subsubsection{Challenge Problems}                                                                                                             
     \Question[10] A builder wishes to place a circular cap of a given height above an existing window. To do this he needs to know the location of 
 the centre of the circle (the cap is not necessarily a semicircle) and the radius of the circle. \(O\) is the centre of the required circle, the   
 radius of the required circle is \(r \mathrm{~cm}\), the width of the window is \(2 d \mathrm{~cm}\) and the height of the circular cap is \(h     
 \mathrm{~cm}\).                                                                                                                                    
     % Image is omitted in the solution                                                                                                             
     \begin{parts}                                                                                                                                  
         \part Express each of these in terms of \(r, d\) and \(h\).                                                                                
         \begin{subparts}                                                                                                                           
             \subpart \(A B\)                                                                                                                       
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(AB = 2d\).                                                                                                                           
         \end{solutionordottedlines}                                                                                                                
             \subpart \(O A\)                                                                                                                       
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(OA = r\).                                                                                                                            
         \end{solutionordottedlines}                                                                                                                
         \end{subparts}                                                                                                                             
         \part Show that \(r=\frac{h^{2}+d^{2}}{2 h}\).                                                                                             
         \begin{solutionordottedlines}[2cm]                                                                                                         
             By the intersecting chords theorem, \(OA^2 = OD \times OB\), where \(OD = r - h\) and \(OB = r + h\). So, \(r^2 = (r - h)(r + h) = r^2 
 h^2\). Adding \(h^2\) to both sides gives \(r^2 + h^2 = 2rh\). Dividing by \(2h\) gives \(r = \frac{h^2 + d^2}{2h}\).                              
         \end{solutionordottedlines}                                                                                                                
         \part If the window is \(120 \mathrm{~cm}\) wide and the cap is \(40 \mathrm{~cm}\) high, find:                                            
         \begin{subparts}                                                                                                                           
             \subpart the radius of the circle                                                                                                      
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(r = \frac{40^2 + 60^2}{2 \times 40} = \frac{1600 + 3600}{80} = \frac{5200}{80} = 65 \mathrm{~cm}\).                                  
         \end{solutionordottedlines}                                                                                                                
             \subpart how far below the top of the window the centre of the circle must be placed                                                   
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(OD = r - h = 65 - 40 = 25 \mathrm{~cm}\).                                                                                            
         \end{solutionordottedlines}                                                                                                                
         \end{subparts}                                                                                                                             
         \part If the builder used a circle of radius \(50 \mathrm{~cm}\) and this produced a cap of height \(20 \mathrm{~cm}\), what was the width 
 of the window?                                                                                                                                     
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(d = \sqrt{2rh - h^2} = \sqrt{2 \times 50 \times 20 - 20^2} = \sqrt{2000 - 400} = \sqrt{1600} = 40 \mathrm{~cm}\). So the width of th 
 window is \(2d = 80 \mathrm{~cm}\).                                                                                                                
         \end{solutionordottedlines}                                                                                                                
     \end{parts}                                                                                                                                    
     \Question[4] A group of \(n\) people attend a club meeting. Before the meeting begins, they all shake hands with each other. Write a formula t 
 find \(H\), the number of handshakes exchanged.                                                                                                    
         \begin{solutionordottedlines}[2cm]                                                                                                         
             Each person shakes hands with \(n - 1\) others, and there are \(n\) people, so there are \(n(n - 1)\) handshakes. However, this counts 
 each handshake twice, so \(H = \frac{n(n - 1)}{2}\).                                                                                               
         \end{solutionordottedlines}                                                                                                                
 \end{questions}                                                                                                                                    
 \subsection{Indices}                                                                                                                               
 \begin{questions}                                                                                                                                  
     \Question[2] Evaluate:                                                                                                                         
     \begin{parts}                                                                                                                                  
         \part \(2^{6}\)                                                                                                                            
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(2^{6} = 64\).                                                                                                                        
         \end{solutionordottedlines}                                                                                                                
         \part \(10^{6}\)                                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(10^{6} = 1,000,000\).                                                                                                                
         \end{solutionordottedlines}                                                                                                                
     \end{parts}                                                                                                                                    
     \Question[1] Express \(120^{2}\) as a product of powers of prime numbers.                                                                      
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(120^{2} = (2^{3} \times 3 \times 5)^{2} = 2^{6} \times 3^{2} \times 5^{2}\).                                                         
         \end{solutionordottedlines}                                                                                                                
     \Question[8] Simplify and evaluate where possible.                                                                                             
     \begin{multicols}{2}\begin{parts}                                                                                                              
         \part \(a^{6} \times a^{7}\)                                                                                                               
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(a^{6} \times a^{7} = a^{6+7} = a^{13}\).                                                                                             
         \end{solutionordottedlines}                                                                                                                
         \part \(2 x^{3} \times 5 x^{6}\)                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(2 x^{3} \times 5 x^{6} = 10 x^{3+6} = 10 x^{9}\).                                                                                    
         \end{solutionordottedlines}                                                                                                                
         \part \(a^{7} \div a^{4}\)                                                                                                                 
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(a^{7} \div a^{4} = a^{7-4} = a^{3}\).                                                                                                
         \end{solutionordottedlines}                                                                                                                
         \part \(\frac{12 b^{7}}{6 b^{2}}\)                                                                                                         
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{12 b^{7}}{6 b^{2}} = 2 b^{7-2} = 2 b^{5}\).                                                                                    
         \end{solutionordottedlines}                                                                                                                
         \part \(\frac{18 p^{10}}{9 p}\)                                                                                                            
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{18 p^{10}}{9 p} = 2 p^{10-1} = 2 p^{9}\).                                                                                      
         \end{solutionordottedlines}                                                                                                                
         \part \(\left(a^{4}\right)^{3}\)                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(a^{4}\right)^{3} = a^{4 \times 3} = a^{12}\).                                                                                  
         \end{solutionordottedlines}                                                                                                                
         \part \(\left(2 a^{7}\right)^{3}\)                                                                                                         
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(2 a^{7}\right)^{3} = 2^{3} a^{7 \times 3} = 8 a^{21}\).                                                                        
         \end{solutionordottedlines}                                                                                                                
         \part \(3 b^{0}\)                                                                                                                          
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(3 b^{0} = 3 \times 1 = 3\).                                                                                                          
         \end{solutionordottedlines}                                                                                                                
     \end{parts}\end{multicols}                                                                                                                     
     \Question[7] Simplify and evaluate where possible.                                                                                             
     \begin{parts}                                                                                                                                  
         \part \[4 a^{2} b^{3} \times 5 a b^{4}\]                                                                                                   
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(4 a^{2} b^{3} \times 5 a b^{4} = 20 a^{2+1} b^{3+4} = 20 a^{3} b^{7}\).                                                              
         \end{solutionordottedlines}                                                                                                                
         \part \[\frac{20 a^{4} b^{2}}{5 a^{2} b}\]                                                                                                 
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{20 a^{4} b^{2}}{5 a^{2} b} = 4 a^{4-2} b^{2-1} = 4 a^{2} b\).                                                                  
         \end{solutionordottedlines}                                                                                                                
         \part \[\frac{24 m^{9} n^{4}}{18 m^{6} n^{2}}\]                                                                                            
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{24 m^{9} n^{4}}{18 m^{6} n^{2}} = \frac{4}{3} m^{9-6} n^{4-2} = \frac{4}{3} m^{3} n^{2}\).                                     
         \end{solutionordottedlines}                                                                                                                
         \part \[\left(3 a^{3} b\right)^{4}\]                                                                                                       
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(3 a^{3} b\right)^{4} = 3^{4} a^{3 \times 4} b^{4} = 81 a^{12} b^{4}\).                                                         
         \end{solutionordottedlines}                                                                                                                
         \part \[\left(5 a^{2} b\right)^{2} \times 4 a^{4} b^{3}\]                                                                                  
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(5 a^{2} b\right)^{2} \times 4 a^{4} b^{3} = 25 a^{4} b^{2} \times 4 a^{4} b^{3} = 100 a^{8} b^{5}\).                           
         \end{solutionordottedlines}                                                                                                                
         \part \[\frac{8 m^{4} n^{2}}{7 m^{3} n} \div \frac{3 m^{3} n^{5}}{14 m^{9} n^{16}}\]                                                       
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{8 m^{4} n^{2}}{7 m^{3} n} \div \frac{3 m^{3} n^{5}}{14 m^{9} n^{16}} = \frac{8 m^{4} n^{2}}{7 m^{3} n} \times \frac{14 m^{9}   
 n^{16}}{3 m^{3} n^{5}} = \frac{8 \times 14 m^{10} n^{17}}{7 \times 3 m^{6} n^{6}} = \frac{112 m^{4} n^{11}}{21} = \frac{16}{3} m^{4} n^{11}\).     
         \end{solutionordottedlines}                                                                                                                
         \part \[\frac{\left(2 x^{2} y\right)^{3}}{5 x^{6} y^{2}} \times\left(\frac{x^{3}}{2 y^{2}}\right)^{3}\]                                    
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\frac{\left(2 x^{2} y\right)^{3}}{5 x^{6} y^{2}} \times\left(\frac{x^{3}}{2 y^{2}}\right)^{3} = \frac{8 x^{6} y^{3}}{5 x^{6} y^{2}}  
 \times \frac{x^{9}}{8 y^{6}} = \frac{8}{5} y \times \frac{x^{9}}{8 y^{6}} = \frac{x^{9}}{5 y^{5}}\).                                               
         \end{solutionordottedlines}                                                                                                                
     \end{parts}                                                                                                                                    
     \Question[6] Evaluate:                                                                                                                         
     \begin{multicols}{2}\begin{parts}                                                                                                              
         \part \(6^{-2}\)                                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(6^{-2} = \frac{1}{6^{2}} = \frac{1}{36}\).                                                                                           
         \end{solutionordottedlines}                                                                                                                
         \part \(8^{-3}\)                                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(8^{-3} = \frac{1}{8^{3}} = \frac{1}{512}\).                                                                                          
         \end{solutionordottedlines}                                                                                                                
         \part \(2^{-7}\)                                                                                                                           
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(2^{-7} = \frac{1}{2^{7}} = \frac{1}{128}\).                                                                                          
         \end{solutionordottedlines}                                                                                                                
         \part \(\left(\frac{4}{5}\right)^{-2}\)                                                                                                    
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(\frac{4}{5}\right)^{-2} = \left(\frac{5}{4}\right)^{2} = \frac{25}{16}\).                                                      
         \end{solutionordottedlines}                                                                                                                
         \part \(\left(\frac{2}{3}\right)^{-4}\)                                                                                                    
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(\frac{2}{3}\right)^{-4} = \left(\frac{3}{2}\right)^{4} = \frac{81}{16}\).                                                      
         \end{solutionordottedlines}                                                                                                                
         \part \(\left(16^{\frac{1}{4}}\right)^{-3}\)                                                                                               
         \begin{solutionordottedlines}[2cm]                                                                                                         
             \(\left(16^{\frac{1}{4}}\right)^{-3} = \left(2\right)^{-3} = \frac{1}{2^{3}} = \frac{1}{8}\).                                          
         \end{solutionordottedlines}                                                                                                                
     \end{parts}\end{multicols}                                                                                                                     
         \Question[4] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(\frac{4 m^{2} n^{5} p^{-6}}{16 m^{-2} n^{5} p^{3}}\)                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{4}{16} \cdot m^{2-(-2)} \cdot n^{5-5} \cdot p^{-6-3}\) \\                                                                                             
                 \(=\frac{1}{4} \cdot m^{4} \cdot p^{-9}\) \\                                                                                                                   
                 \(=\frac{m^{4}}{4p^{9}}\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(2^{2} y^{3}\right)^{-5}\)                                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(2^{2 \cdot -5}) \cdot (y^{3 \cdot -5})\) \\                                                                                                                
                 \(=2^{-10} \cdot y^{-15}\) \\                                                                                                                                  
                 \(=\frac{1}{2^{10} y^{15}}\)                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(5^{-2} x^{3}\right)^{-5}\)                                                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(5^{-2 \cdot -5}) \cdot (x^{3 \cdot -5})\) \\                                                                                                               
                 \(=5^{10} \cdot x^{-15}\) \\                                                                                                                                   
                 \(=\frac{5^{10}}{x^{15}}\)                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(3^{-3} a^{2} b^{-1}\right)^{-4}\)                                                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(3^{-3 \cdot -4}) \cdot (a^{2 \cdot -4}) \cdot (b^{-1 \cdot -4})\) \\                                                                                       
                 \(=3^{12} \cdot a^{-8} \cdot b^{4}\) \\                                                                                                                        
                 \(=\frac{3^{12} b^{4}}{a^{8}}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(4 a^{2} \times 5 a^{-3}\)                                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4 \cdot 5 \cdot a^{2-3}\) \\                                                                                                                                
                 \(=20 a^{-1}\) \\                                                                                                                                              
                 \(=\frac{20}{a}\)                                                                                                                                              
             \end{solutionordottedlines}                                                                                                                                        
             \part \(14 a^{-4} \div 7 a^{-5}\)                                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{14}{7} \cdot a^{-4-(-5)}\) \\                                                                                                                         
                 \(=2 a^{1}\) \\                                                                                                                                                
                 \(=2a\)                                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\frac{2 m^{3} n^{4}}{(5 m)^{2}} \times \frac{10 m}{3 n^{-4}}\)                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{2 m^{3} n^{4}}{25 m^{2}} \cdot \frac{10 m}{3 n^{-4}}\) \\                                                                                             
                 \(=\frac{2}{25} \cdot \frac{10}{3} \cdot m^{3-2+1} \cdot n^{4+4}\) \\                                                                                          
                 \(=\frac{20}{75} \cdot m^{2} \cdot n^{8}\) \\                                                                                                                  
                 \(=\frac{4}{15} m^{2} n^{8}\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Evaluate:                                                                                                                                                 
         \begin{parts}                                                                                                                                                          
             \part \(49^{\frac{1}{2}}\)                                                                                                                                         
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\sqrt{49}\) \\                                                                                                                                              
                 \(=7\)                                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part \(125^{\frac{2}{3}}\)                                                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\sqrt[3]{125^{2}}\) \\                                                                                                                                      
                 \(=\sqrt[3]{15625}\) \\                                                                                                                                        
                 \(=25\)                                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(\frac{1}{8}\right)^{-\frac{2}{3}}\)                                                                                                                  
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=(8^{\frac{2}{3}})\) \\                                                                                                                                      
                 \(=\sqrt[3]{8^{2}}\) \\                                                                                                                                        
                 \(=\sqrt[3]{64}\) \\                                                                                                                                           
                 \(=4\)                                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[4] Simplify, expressing the answer with positive indices.                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(3 b^{\frac{2}{3}} \times 4 b\)                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3 \cdot 4 \cdot b^{\frac{2}{3}+1}\) \\                                                                                                                      
                 \(=12 b^{\frac{5}{3}}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part \(p^{\frac{2}{3}} \div p^{\frac{1}{2}}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=p^{\frac{2}{3}-\frac{1}{2}}\) \\                                                                                                                            
                 \(=p^{\frac{4}{6}-\frac{3}{6}}\) \\                                                                                                                            
                 \(=p^{\frac{1}{6}}\)                                                                                                                                           
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(2 x^{-\frac{1}{3}}\right)^{-2}\)                                                                                                                     
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=2^{-2} \cdot x^{\frac{2}{3}}\) \\                                                                                                                           
                 \(=\frac{1}{4} x^{\frac{2}{3}}\)                                                                                                                               
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\left(8 p^{-2} q^{3}\right)^{\frac{1}{2}}\)                                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=8^{\frac{1}{2}} \cdot p^{-1} \cdot q^{\frac{3}{2}}\) \\                                                                                                     
                 \(=2 \cdot \frac{q^{\frac{3}{2}}}{p}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Write in scientific notation.                                                                                                                             
         \begin{parts}                                                                                                                                                          
             \part 164000000                                                                                                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.64 \times 10^{8}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.0047                                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.7 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.0035                                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3.5 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[3] Write in decimal form.                                                                                                                                    
         \begin{parts}                                                                                                                                                          
             \part \(6.8 \times 10^{4}\)                                                                                                                                        
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=68000\)                                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(9.4 \times 10^{-2}\)                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=0.094\)                                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                                        
             \part \(3.2 \times 10^{-4}\)                                                                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=0.00032\)                                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[2] Simplify, writing each answer in scientific notation.                                                                                                     
         \begin{parts}                                                                                                                                                          
             \part \(\left(3.1 \times 10^{4}\right) \times\left(2 \times 10^{-2}\right)\)                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=3.1 \cdot 2 \times 10^{4-2}\) \\                                                                                                                            
                 \(=6.2 \times 10^{2}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part \(\frac{\left(3 \times 10^{4}\right)^{3}}{9 \times 10^{-2}}\)                                                                                                
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=\frac{3^{3} \times 10^{12}}{9 \times 10^{-2}}\) \\                                                                                                          
                 \(=\frac{27}{9} \times 10^{12+2}\) \\                                                                                                                          
                 \(=3 \times 10^{14}\)                                                                                                                                          
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}                                                                                                                                                            
         \Question[6] Write in scientific notation correct to the number of significant figures indicated in the brackets.                                                      
         \begin{multicols}{2}\begin{parts}                                                                                                                                      
             \part 18.62\hfill{}(2)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.9 \times 10^{1}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part 18.62\hfill{}(3)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=1.86 \times 10^{1}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 18.62\hfill{}(1)                                                                                                                                             
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=2 \times 10^{1}\)                                                                                                                                           
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.004276\hfill{}(2)                                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.3 \times 10^{-3}\)                                                                                                                                        
             \end{solutionordottedlines}                                                                                                                                        
             \part 5973.4\hfill{}(2)                                                                                                                                            
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=6.0 \times 10^{3}\)                                                                                                                                         
             \end{solutionordottedlines}                                                                                                                                        
             \part 0.473952\hfill{}(3)                                                                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                                                                 
                 \(=4.74 \times 10^{-1}\)                                                                                                                                       
             \end{solutionordottedlines}                                                                                                                                        
         \end{parts}\end{multicols}                                                                                                                                             
     \end{questions}                                                                                                                                                            
 \end{problembox}                                                                                                                                                               
                                                                                                                                                                                
