Let us begin simply with the following:

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[1] Express as a power or as a product of powers.
        \begin{parts}
            \part \(5 \times 5 \times 5\)
            \begin{solutionordottedlines}
                \(5 \times 5 \times 5=5^{3} \quad\) 
            \end{solutionordottedlines}
            \part \(3 \times 3 \times 7 \times 7 \times 7 \times 7\)
            \begin{solutionordottedlines}
                \(3 \times 3 \times 7 \times 7 \times 7 \times 7=3^{2} \times 7^{4}\)
            \end{solutionordottedlines}
            \part
        \end{parts}
        \Question[1] Express each number as a power of a prime.
        \begin{parts}
            \part 81
            \begin{solutionordottedlines}[2cm]
                \(\quad 81=3 \times 3 \times 3 \times 3\)
            \end{solutionordottedlines}
            \part 128
            \begin{solutionordottedlines}[2cm]
                \(128=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)
                \(=3^{4}\)
                \(=2^{7}\)
            \end{solutionordottedlines}
        \end{parts}
    \end{questions}
\end{examplebox}

Now some definitions:
\begin{boxdef}
    \textbf{Base:}
    \begin{solutionordottedlines}[1cm]
        The number 2 in \(2^{4}\) is called the base.
    \end{solutionordottedlines}
\end{boxdef}
\begin{boxdef}
    \textbf{Index:}
    \begin{solutionordottedlines}[1cm]
        The number 4 in \(2^{4}\) is called the index or exponent.
    \end{solutionordottedlines}
\end{boxdef}

\begin{boxlaw}
    \subsection*{Index Laws}
    \subsubsection*{Product of Powers}
        To multiply powers of the same base, add the indices.

        \[
        a^{m} a^{n}=a^{m+n}
        \]

    \subsubsection*{Division of Powers}
        To divide powers of the same base, subtract the indices.

        \[
        \frac{a^{m}}{a^{n}}=a^{m-n} \quad \text { where } m>n \text { and } a \neq 0
        \]
    \subsubsection*{Powers of Powers}
        To raise a power to a power, multiply the indices.

        \[
        \left(a^{m}\right)^{n}=a^{m n}
        \]
    \subsubsection*{Power of Products}

        A power of a product is the product of the powers.

        \[(a b)^{m}=a^{m} b^{m}\]

    \subsubsection*{Power of Quotient}
        A power of a quotient is the quotient of the powers.

        \[
        \left(\frac{a}{b}\right)^{m}=\frac{a^{m}}{b^{m}} \quad \text { where } b \neq 0
        \]
    \subsubsection*{Zero Exponent Law}
        Any base raised to the power of $0$ is equal to $1$.
        \[a^0 = 1\]
    \subsubsection*{Negative Exponent Law (next lesson)}
         \[ a^{-n} = \frac{1}{a^n} \] 
         A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent.
\end{boxlaw}

Now with these laws in mind, try the following 10 examples:
\begin{examplebox}
    \subsection{Examples}
    \begin{questions}
        \Question[10] Simplify, expressing the answer in index form.
        \begin{parts}\begin{multicols}{2}
            \part \(3^{2} \times 3^{4}\)
                \begin{solutionordottedlines}[2cm]
                    \(3^{2} \times 3^{4}=3^{6}\)
                \end{solutionordottedlines}
            \part \(a^{3} \times a^{5}\)
                \begin{solutionordottedlines}[2cm]
                    \(a^{3} \times a^{5}=a^{8}\)
                \end{solutionordottedlines}
            \part \(3 x^{2} \times x^{3}\)
                \begin{solutionordottedlines}[2cm]
                    \(3 x^{2} \times x^{3}=3 x^{5}\)
                \end{solutionordottedlines}
            \part \(2 a^{2} b^{3} \times 5 a b^{2}\)
                \begin{solutionordottedlines}[2cm]
                    \(2 a^{2} b^{3} \times 5 a b^{2}=10 \times a^{2+1} \times b^{3+2}\)

                    \[
                    =10 a^{3} b^{5}
                    \]
                \end{solutionordottedlines}
            \part \(\frac{3^{5}}{3^{2}}\)
                \begin{solutionordottedlines}[2cm]
                    \(\frac{3^{5}}{3^{2}}=3^{5-2}\)
                \end{solutionordottedlines}
            \part \(\frac{9^{5}}{9^{4}}\)
                \begin{solutionordottedlines}[2cm]
                    \(\frac{9^{5}}{9^{4}}=9^{1}\)
                \end{solutionordottedlines}
            \part \(10^{6} \div 10^{4}\)
                \begin{solutionordottedlines}[2cm]
                    \(10^{6} \div 10^{4}=10^{6-4}\)
                    \(=3^{3}\)
                    \(=9\)
                    \(=10^{2}\)
                    \(=27\)
                    \(=100\)
                \end{solutionordottedlines}
            \part \(a^{7} \div a^{4}\)
                \begin{solutionordottedlines}[2cm]
                    \(\begin{aligned} a^{7} \div a^{4} & =a^{7-4} \\ & =a^{3}\end{aligned}\)
                \end{solutionordottedlines}
            \part \(\frac{3 y^{4}}{y}\)
                \begin{solutionordottedlines}[2cm]
                    \(\frac{3 y^{4}}{y}=3 \times \frac{y^{4}}{y}\)
                \end{solutionordottedlines}
            \part \(\frac{6 x^{5}}{2 x^{3}}\)
                \begin{solutionordottedlines}[2cm]
                    \(\frac{6 x^{5}}{2 x^{3}}=\frac{6}{2} \times \frac{x^{5}}{x^{3}}\)
                    \(=3 \times y^{4-1}\)
                    \(=3 \times x^{5-3}\)
                    \(=3 y^{3}\)
                    \(=3 x^{2}\)
                \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \Question[4] Double marks for each of these:
        \begin{parts}
            \part \[\frac{3 x^{3} y^{2}}{4 x y} \times \frac{6 x^{2} y^{3}}{x^{3} y^{2}}\]
            \begin{solutionordottedlines}[2cm]
                \(\frac{3 x^{3} y^{2}}{4 x y} \times \frac{6 x^{2} y^{3}}{x^{3} y^{2}}=\frac{18 x^{5} y^{5}}{4 x^{4} y^{3}}\)
                \(=\frac{9 x y^{2}}{2}\)
            \end{solutionordottedlines}
            \part \[\frac{8 a^{2} b^{3}}{3 a^{3} b} \div \frac{4 a b^{2}}{9 a^{3} b^{5}}\]
            \begin{solutionordottedlines}[2cm]
                \(\frac{8 a^{2} b^{3}}{3 a^{3} b} \div \frac{4 a b^{2}}{9 a^{3} b^{5}}=\frac{8 a^{2} b^{3}}{3 a^{3} b} \times \frac{9 a^{3} b^{5}}{4 a b^{2}}\)

                \[
                =\frac{72 \times a^{5} \times b^{8}}{12 \times a^{4} \times b^{3}}
                \]

                \[
                =6 a b^{5}
                \]
            \end{solutionordottedlines}
        \end{parts}
        \Question[6] And a $1.5\times$ multiplier for these:
        \begin{parts}
            \part \(\left(\frac{2}{3}\right)^{2}\)
            \begin{solutionordottedlines}[2cm]
                \(\begin{aligned}\left(\frac{2}{3}\right)^{3} & =\frac{2^{3}}{3^{3}} \\ & =\frac{8}{27}\end{aligned}\)
            \end{solutionordottedlines}
            \part \(\left(\frac{m}{n}\right)^{5}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(\frac{m}{n}\right)^{5}=\frac{m^{5}}{n^{5}}\)
            \end{solutionordottedlines}
            \part \(\left(\frac{x^{3}}{y^{2}}\right)^{2} \times\left(\frac{y}{x}\right)^{4}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(\frac{x^{3}}{y^{2}}\right)^{2} \times\left(\frac{y}{x}\right)^{4}=\frac{x^{6}}{y^{4}} \times \frac{y^{4}}{x^{4}}\)
            \end{solutionordottedlines}
            \part \(\left(\frac{2 x^{2}}{3}\right)^{2} \div \frac{4 x^{3}}{9}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(\frac{2 x^{2}}{3}\right)^{2} \div \frac{4 x^{3}}{9}=\frac{4 x^{4}}{9} \times \frac{9}{4 x^{3}}\)
                \(=x^{2}\)
                \(=x\)
            \end{solutionordottedlines}
        \end{parts}
        \begin{parts}
            \part \(\left(5 a^{3}\right)^{0}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(5 a^{3}\right)^{0}=1\)
            \end{solutionordottedlines}
            \part \(\frac{6 x^{2} y}{x y^{2}} \times \frac{y^{3} x}{2 y^{2} x^{2}}\)
            \begin{solutionordottedlines}[2cm]
                \[
                \begin{aligned}
                \frac{6 x^{2} y}{x y^{2}} \times \frac{y^{3} x}{2 y^{2} x^{2}} & =\frac{6}{2} \times \frac{x^{3}}{x^{3}} \times \frac{y^{4}}{y^{4}} \\
                & =3 x^{0} y^{0} \\
                & =3 \times 1 \times 1 \\
                & =3
                \end{aligned}
                \]
            \end{solutionordottedlines}
            \part \(\left(m n^{2}\right)^{0}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(m n^{2}\right)^{0}=1\)
            \end{solutionordottedlines}
            \part \(\left(a^{4} b^{2}\right)^{3}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(a^{4} b^{2}\right)^{3}=a^{12} b^{6}\)
            \end{solutionordottedlines}
            \part \(\left(2 a^{4}\right)^{3}\)
            \begin{solutionordottedlines}[2cm]
                \(\left(2 a^{4}\right)^{3}=2^{3} \times a^{12}\)
            \end{solutionordottedlines}
            \part \(2\left(x^{2} y\right)^{0} \times\left(x^{2} y^{3}\right)^{3}\)
            \begin{solutionordottedlines}[2cm]
                \(2\left(x^{2} y\right)^{0} \times\left(x^{2} y^{3}\right)^{3}=2 \times 1 \times x^{6} y^{9}\)
                \(=8 a^{12}\)
                \(=2 x^{6} y^{9}\)
            \end{solutionordottedlines}
            \part
        \end{parts}
    \end{questions}
\end{examplebox}

\fbox{\fbox{\parbox{\textwidth}{\textbf{Note: }There are different possible interpretations of the word 'simplify'. There may be more than one acceptable simplified form.
}}}

                                                                                                                                                            
 \begin{exercisebox}                                                                                                                                        
     \subsection{Exercises:}                                                                                                                                
     \begin{questions}                                                                                                                                      
         \Question[3] State the base and index of:                                                                                                          
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(6^{4}=\fillin[6]\)                                                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 Base: 6, Index: 4                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(7^{3}=\fillin[7]\)                                                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 Base: 7, Index: 3                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(8^{2}=\fillin[8]\)                                                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 Base: 8, Index: 2                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Express as a power of a prime number.                                                                                                 
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part $8 = \fillin[$2^3$]$                                                                                                                       
             \part $27 = \fillin[$3^3$]$                                                                                                                      
             \part $64 = \fillin[$2^6$]$                                                                                                                      
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Evaluate:                                                                                                                             
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(3^{4}=\fillin[81]\)                                                                                                                    
             \part \(2^{7}=\fillin[128]\)                                                                                                                   
             \part \(5^{5}=\fillin[3125]\)                                                                                                                  
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Express as a product of powers of prime numbers.                                                                                      
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part $18=\fillin[$2 \times 3^2$]$                                                                                                               
             \part $24=\fillin[$2^3 \times 3$]$                                                                                                               
             \part $144=\fillin[$2^4 \times 3^2$]$                                                                                                            
         \end{multicols}\end{parts}                                                                                                                         
         \Question[3] Simplify:                                                                                                                             
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(2^{7} \times 2^{3}\)                                                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(2^{10}\)                                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(3^{3} \times 3^{4} \times 3^{5}\)                                                                                                      
             \begin{solutionordottedlines}                                                                                                                  
                 \(3^{12}\)                                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(3 x^{2} \times 4 x^{3}\)                                                                                                               
             \begin{solutionordottedlines}                                                                                                                  
                 \(12 x^{5}\)                                                                                                                               
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] Simplify:                                                                                                                             
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(a^{2} b^{3} \times b^{2}\)                                                                                                             
             \begin{solutionordottedlines}                                                                                                                  
                 \(a^{2} b^{5}\)                                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(a^{3} b \times a^{2} b^{3}\)                                                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(a^{5} b^{4}\)                                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(2 x y^{2} \times 3 x^{2} y\)                                                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(6 x^{3} y^{3}\)                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(4 a^{3} b^{2} \times a^{2} b^{4}\)                                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 \(4 a^{5} b^{6}\)                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[6]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{3^7}{3^2}\)                                                                                                                      
             \begin{solutionordottedlines}                                                                                                                  
                 \(3^5\)                                                                                                                                    
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{2^{6}}{2^{2}}\)                                                                                                                  
             \begin{solutionordottedlines}                                                                                                                  
                 \(2^4\)                                                                                                                                    
             \end{solutionordottedlines}                                                                                                                    
             \part \(10^{7} \div 10^{2}\)                                                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(10^5\)                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{10^{12}}{10^{4}}\)                                                                                                               
             \begin{solutionordottedlines}                                                                                                                  
                 \(10^8\)                                                                                                                                   
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{2 x^{3}}{x^{2}}\)                                                                                                                
             \begin{solutionordottedlines}                                                                                                                  
                 \(2 x\)                                                                                                                                    
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{6 x^{5}}{2 x^{2}}\)                                                                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 x^{3}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{a^{3} b^{2}}{a^{2}}\)                                                                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(a b^{2}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{x^{3} y^{2}}{x y}\)                                                                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(x^{2} y\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{12 a^{6} b^{2}}{4 a^{2} b}\)                                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 a^{4} b\)                                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{15 x y^{3}}{3 y^{2}}\)                                                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(5 x y\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\frac{a^{3} b^{2}}{a b} \times \frac{a^{2} b}{a}\)                                                                                     
             \begin{solutionordottedlines}                                                                                                                  
                 \(a^{4} b^{2}\)                                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{x^{3} y}{x y^{2}} \times \frac{x^{4} y^{5}}{x^{2}}\)                                                                             
             \begin{solutionordottedlines}                                                                                                                  
                 \(x^{4} y^{4}\)                                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{6 a b^{2}}{5 a^{3} b} \div \frac{12 a b}{15 a^{5} b}\)                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 a^{2} b\)                                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{7 x^{3} y^{4}}{2 x y^{2}} \div \frac{21 x^{2} y^{3}}{4 x^{3} y^{2}}\)                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(2 x^{2} y\)                                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[6]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(a^{4} \times \fillin[$a^{6}$]=a^{10}\)                                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(a^{6}\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
             \part \(9 d^{5} \times \fillin[3]=27 d^{6}\)                                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 d\)                                                                                                                                    
             \end{solutionordottedlines}                                                                                                                    
             \part \(15 d^{7} \div \fillin[$5 d^{5}$]=3 d^{2}\)                                                                                               
             \begin{solutionordottedlines}                                                                                                                  
                 \(5 d^{5}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \(8 a b^{4} \times \fillin[$3 a b^{2}$]=24 a^{2} b^{6}\)                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(3 a b^{2}\)                                                                                                                              
             \end{solutionordottedlines}                                                                                                                    
             \part \(\ell^{6} m^{7} \div \fillin[$\ell^{4} m^{2}$]=\ell^{2} m^{5}\)                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(\ell^{4} m^{2}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                    
             \part \(b^{7} \times \fillin[$b^{9}$]=b^{16}\)                                                                                                   
             \begin{solutionordottedlines}                                                                                                                  
                 \(b^{9}\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[2]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(a^{0}\)                                                                                                                                
             \begin{solutionordottedlines}                                                                                                                  
                 \(1\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(2 x^{0}\)                                                                                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(2\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[6]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{3}                                                                                                                  
             \part \(3 a^{0}\)                                                                                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(3\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(6 a^{0}\)                                                                                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(6\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(4 a^{0}+3 b^{0}\)                                                                                                                      
             \begin{solutionordottedlines}                                                                                                                  
                 \(7\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \(6 a^{0}+7 m^{0}\)                                                                                                                      
             \begin{solutionordottedlines}                                                                                                                  
                 \(13\)                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                    
             \part \((4 b)^{0}+2 b^{0}\)                                                                                                                    
             \begin{solutionordottedlines}                                                                                                                  
                 \(3\)                                                                                                                                      
             \end{solutionordottedlines}                                                                                                                    
             \part \((3 b)^{0}-5 d^{0}\)                                                                                                                    
             \begin{solutionordottedlines}                                                                                                                  
                 \(-4\)                                                                                                                                     
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[2] Simplify, leaving the answer as a power.                                                                                              
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(2^{3}\right)^{4}\)                                                                                                               
             \begin{solutionordottedlines}                                                                                                                  
                 \(2^{12}\)                                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(3^{2}\right)^{3}\)                                                                                                               
             \begin{solutionordottedlines}                                                                                                                  
                 \(3^{6}\)                                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4] $\,$                                                                                                                                  
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(a^{3}\right)^{2} \times\left(a^{3}\right)^{4}\)                                                                                  
             \begin{solutionordottedlines}                                                                                                                  
                 \(a^{14}\)                                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(x^{4}\right)^{2} \times\left(x^{3}\right)^{3}\)                                                                                  
             \begin{solutionordottedlines}                                                                                                                  
                 \(x^{17}\)                                                                                                                                 
             \end{solutionordottedlines}                                                                                                                    
             \part \(2 a b^{2} \times 3 a\left(b^{3}\right)^{2}\)                                                                                           
             \begin{solutionordottedlines}                                                                                                                  
                 \(6 a^{2} b^{8}\)                                                                                                                          
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{3 a b}{\left(b^{2}\right)^{3}} \times \frac{4 b^{7}}{3 a}\)                                                                      
             \begin{solutionordottedlines}                                                                                                                  
                 \(4 b^{2}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \((3 a)^{2}\)                                                                                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(9 a^{2}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \((2 x)^{3}\)                                                                                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(8 x^{3}\)                                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{a}{5}\right)^{2}\)                                                                                                         
             \begin{solutionordottedlines}                                                                                                                  
                 \(\frac{a^{2}}{25}\)                                                                                                                       
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{2}{x}\right)^{3}\)                                                                                                         
             \begin{solutionordottedlines}                                                                                                                  
                 \(\frac{8}{x^{3}}\)                                                                                                                        
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(2 a^{2} b\right)^{2} \times 3 a b^{3}\)                                                                                          
             \begin{solutionordottedlines}                                                                                                                  
                 \(12 a^{5} b^{5}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(3 x y^{2}\right)^{3} \times\left(x^{2} y\right)^{2}\)                                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(27 x^{5} y^{8}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(5 x y^{2}\right)^{3} \times\left(x^{2} y^{3}\right)^{2}\)                                                                        
             \begin{solutionordottedlines}                                                                                                                  
                 \(125 x^{7} y^{13}\)                                                                                                                       
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(2 a^{3} b\right)^{3} \times 3 a^{0}\)                                                                                            
             \begin{solutionordottedlines}                                                                                                                  
                 \(24 a^{9} b^{3}\)                                                                                                                         
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
         \Question[4]$\,$                                                                                                                                   
         \begin{parts}\begin{multicols}{2}                                                                                                                  
             \part \(\left(\frac{x^{2}}{y}\right)^{2} \times\left(\frac{y^{2}}{x}\right)^{3}\)                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(\frac{x y^{4}}{y^{2}}\)                                                                                                                  
             \end{solutionordottedlines}                                                                                                                    
             \part \(\left(\frac{4 a^{2}}{b}\right)^{2} \times\left(\frac{b}{2 a}\right)^{3}\)                                                              
             \begin{solutionordottedlines}                                                                                                                  
                 \(\frac{8 a b}{b^{2}}\)                                                                                                                    
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{\left(3 x y^{2}\right)^{2} \times\left(2 x^{2} y\right)^{3}}{\left(6 x^{2} y\right)^{2}}\)                                       
             \begin{solutionordottedlines}                                                                                                                  
                 \(x^{2} y^{4}\)                                                                                                                            
             \end{solutionordottedlines}                                                                                                                    
             \part \(\frac{3 a^{2} b^{4} \times\left(2 a b^{2}\right)^{3}}{\left(4 a^{2} b^{3}\right)^{2}}\)                                                
             \begin{solutionordottedlines}                                                                                                                  
                 \(\frac{3 a^{2} b^{2}}{4}\)                                                                                                                
             \end{solutionordottedlines}                                                                                                                    
         \end{multicols}\end{parts}                                                                                                                         
     \end{questions}                                                                                                                                        
 \end{exercisebox}