Up until now we have been dealing with \emph{whole} numbers in our powers. We have been manipulating expressions such as \(a^n\) where $n \in \mathbb{Z}$. But truthfully, there is no reason why we must confine ourselves to $\mathbb{Z}$, we have the tools to do arithmetic with $\mathbb{Q}$, \fillin[the set of all quotients][2in]. 

Later, if you fall in the black hole of mathematics, you will learn to grasp $a$ to the power of \emph{all real} numbers $\mathbb{R}$, things such as \(2^\pi\) and later even complex numbers (denoted $\mathbb{C}$) to understand the most beautiful equation in mathematics (by votes): \[e^{\pi i} = -1\]

But for now we will just understand the \emph{quotient} powers thoroughly.

There are 3 main types:
\begin{enumerate}
\begin{centering}
    \item numerator of one: $2^\frac{1}{3}$
    \item numerator of not one: $2^\frac{2}{3}$
    \item negative versions of the above cases $\uparrow$
\end{centering}
\end{enumerate}


\begin{derivebox}%TODO: update the boxstyle
    You could just remember that $\sqrt{a} = a^\frac{1}{2}$. Or you could \emph{understand} that this is obviously true from the arithmetic laws you learnt last lesson.

    Recall that $(a^2)^2 = a^4$, then $a^1 = (a^\frac{1}{2})^2$ which we could say is the same as $a^\frac{1}{2} \times a^\frac{1}{2}$. But then which mathematical entity becomes itself when squared?

    \begin{solutionorbox}
        $\sqrt{a}$
    \end{solutionorbox}

    Thus, obviously $a^\frac{1}{2} = \fillin[$\sqrt{a}$]$
\end{derivebox}


\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[3]$\,$
        \begin{parts}\begin{multicols}{3}
            \part $100^\frac{1}{2} = \fillin[10]$
            \part $4^\frac{1}{2} = \fillin[2]$
            \part $256^\frac{1}{2} = \fillin[16]$
        \end{multicols}\end{parts}
    \end{questions}
\end{examplebox}

\begin{derivebox}
    We can extend this more generally, where the denominator does not need to be 2. Recall that $\sqrt[3]{8} = \fillin[2]$.\\

    And so $8^\frac{1}{3} = 2$ and $a^\frac{1}{n} = \sqrt[n]{a}$. 
\end{derivebox}

\begin{examplebox}
    \subsection{Examples}
    \begin{questions}
        \Question[6]$\,$
        \begin{multicols}{3}
            \begin{parts}
                \part $\sqrt[3]{27} = \fillin[3]$
                \part $\sqrt[4]{16} = \fillin[2]$
                \part $\sqrt[3]{125} = \fillin[2]$
            \end{parts}
            \begin{parts}
                \part $9^\frac{1}{2} = \fillin[3]$
                \part $27^\frac{1}{3} = \fillin[3]$
                \part $16^\frac{1}{4} = \fillin[2]$
            \end{parts}
        \end{multicols}
    \end{questions}
\end{examplebox}

\begin{derivebox}
    Notice that up until now all of the numerators have been the number 1. It is easy to equip ourselves to handle bigger numbers though. Just recall the law from last lesson: \[(a^m)^n = a^mn\]

    Which we can apply to the \(a^\frac{1}{n}\)'s we have by raising this to whatever number we need. Thus \(a^\frac{2}{3} = (a^\frac{1}{3})^2\).

    We can even change the order around to be $(a^2)^\frac{1}{3}$ which then looks nicer as $\sqrt[3]{a^2}$.

    \begin{theorembox}
        \subsection*{Positive fractional indices}
        \[a^\frac{p}{q} = (a^\frac{1}{q})^p = (\sqrt[q]{a})^p\]
    \end{theorembox}
\end{derivebox}

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[3]$\,$
        \begin{multicols}{3}\begin{parts}
            \part $4^\frac{3}{2} = \fillin[8]$
            \part $8^\frac{2}{3} = \fillin[4]$
            \part $81^\frac{3}{4} = \fillin[27]$
        \end{parts}\end{multicols}
    \end{questions}
\end{examplebox}

\begin{derivebox}
    The final thing in this section to learn is the \emph{negative fractional indices}. Recall that \[a^{-m} = \frac{1}{a^m}\].

Then in exactly the same fashion, \[a^{-\frac{p}{q}} = \frac{1}{a^\frac{p}{q}}\]
\end{derivebox}

\begin{examplebox}
    \subsection{Examples:}
    \begin{questions}
        \Question[2]$\,$
        \begin{multicols}{2}\begin{parts}
            \part $4^{-\frac{3}{2}}=$
            \begin{solutionordottedlines}
                $\frac{1}{8}$
            \end{solutionordottedlines}
            \part $(\frac{8}{27})^{-\frac{1}{3}}=$
            \begin{solutionordottedlines}
                $\frac{3}{2}$
            \end{solutionordottedlines}
        \end{parts}\end{multicols}
    \end{questions}
\end{examplebox}

 \begin{exercisebox}                                                                                            
     \subsection{Exercises:}                                                                                    
     \begin{questions}                                                                                          
         \Question[3] Evaluate:                                                                                 
         \begin{parts}                                                                                          
             \begin{multicols}{3}                                                                               
                 \part \(\sqrt[3]{8}\)                                                                          
                 \begin{solutionordottedlines}[2cm]                                                             
                     2                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(\sqrt[5]{32}\)                                                                         
                 \begin{solutionordottedlines}[2cm]                                                             
                     2                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(\sqrt[3]{216}\)                                                                        
                 \begin{solutionordottedlines}[2cm]                                                             
                     6                                                                                          
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[3] Write using fractional indices. Evaluate, correct to 4 decimal places.                    
         \begin{parts}                                                                                          
             \begin{multicols}{3}                                                                               
                 \part \(\sqrt{14}\)                                                                            
                 \begin{solutionordottedlines}[2cm]                                                             
                    \(14^{\frac{1}{2}} \approxeq 3.7417\)
                 \end{solutionordottedlines}                                                                    
                 \part \(\sqrt[4]{64}\)                                                                         
                 \begin{solutionordottedlines}[2cm]                                                             
                    \(64^{\frac{1}{4}} = 2.8284\)
                 \end{solutionordottedlines}                                                                    
                 \part \(\sqrt[5]{7}\)                                                                          
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(7^{\frac{1}{5}} \approxeq 1.4758\)                                                             
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[6] Evaluate:                                                                                 
         \begin{parts}                                                                                          
             \begin{multicols}{3}                                                                               
                 \part \(4^{\frac{1}{2}}\)                                                                      
                 \begin{solutionordottedlines}[2cm]                                                             
                     2                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(27^{\frac{1}{3}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     3                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(64^{\frac{1}{2}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     8                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(25^{\frac{1}{2}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     5                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(32^{\frac{1}{5}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     2                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(625^{\frac{1}{4}}\)                                                                    
                 \begin{solutionordottedlines}[2cm]                                                             
                     5                                                                                          
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[6] Evaluate:                                                                                 
         \begin{parts}                                                                                          
             \begin{multicols}{3}                                                                               
                 \part \(4^{\frac{5}{2}}\)                                                                      
                 \begin{solutionordottedlines}[2cm]                                                             
                     32                                                                                         
                 \end{solutionordottedlines}                                                                    
                 \part \(25^{\frac{3}{2}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     125                                                                                        
                 \end{solutionordottedlines}                                                                    
                 \part \(32^{\frac{2}{5}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     4                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(81^{\frac{3}{4}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     27                                                                                         
                 \end{solutionordottedlines}                                                                    
                 \part \((\sqrt[4]{16})^{3}\)                                                                   
                 \begin{solutionordottedlines}[2cm]                                                             
                     8                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \((\sqrt[3]{27})^{2}\)                                                                   
                 \begin{solutionordottedlines}[2cm]                                                             
                     9                                                                                          
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[8] Simplify:                                                                                 
         \begin{parts}                                                                                          
             \begin{multicols}{2}                                                                               
                 \part \(\left(a^{\frac{1}{2}}\right)^{2}\)                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     a                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(\left(b^{\frac{1}{3}}\right)^{6}\)                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(b^2\)                                                                                        
                 \end{solutionordottedlines}                                                                    
                 \part \(x^{\frac{1}{2}} \times x^{\frac{3}{2}}\)                                               
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(x^2\)                                                                                        
                 \end{solutionordottedlines}                                                                    
                 \part \(y^{\frac{1}{3}} \times y^{\frac{2}{3}}\)                                               
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(y\)                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(x^{\frac{3}{2}} \div x^{\frac{1}{2}}\)                                                 
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(x\)                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(y^{\frac{2}{3}} \div y^{\frac{1}{3}}\)                                                 
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(y^{\frac{1}{3}}\)                                                                            
                 \end{solutionordottedlines}                                                                    
                 \part \(\left(4 m^{6}\right)^{\frac{1}{2}}\)                                                   
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(2m^3\)                                                                                       
                 \end{solutionordottedlines}                                                                    
                 \part \(\left(27 n^{12}\right)^{\frac{1}{3}}\)                                                 
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(3n^4\)                                                                                       
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[4] Evaluate:                                                                                 
         \begin{parts}                                                                                          
             \begin{multicols}{2}                                                                               
                 \part \(4^{-\frac{1}{2}}\)                                                                     
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(\frac{1}{2}\)                                                                                
                 \end{solutionordottedlines}                                                                    
                 \part \(25^{-\frac{1}{2}}\)                                                                    
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(\frac{1}{5}\)                                                                                
                 \end{solutionordottedlines}                                                                    
                 \part \(\left(\frac{1}{81}\right)^{-\frac{1}{4}}\)                                             
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(3\)                                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(81^{-\frac{1}{4}}\)                                                                    
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(\frac{1}{3}\)                                                                                
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
         \Question[6] Simplify, expressing the answer with positive indices.                                    
         \begin{parts}                                                                                          
             \begin{multicols}{3}                                                                               
                 \part \(\left(a^{\frac{1}{2}}\right)^{-2}\)                                                    
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(a^{-1}\)                                                                                     
                 \end{solutionordottedlines}                                                                    
                 \part \(\left(b^{-\frac{2}{3}}\right)^{6}\)                                                    
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(b^{-4}\)                                                                                     
                 \end{solutionordottedlines}                                                                    
                 \part \(y^{\frac{1}{3}} \times y^{-\frac{2}{3}}\)                                              
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(y^{-\frac{1}{3}}\)                                                                           
                 \end{solutionordottedlines}                                                                    
                 \part \(p^{\frac{3}{4}} \times p^{-\frac{2}{5}}\)                                              
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(p^{\frac{7}{20}}\)                                                                           
                 \end{solutionordottedlines}                                                                    
                 \part \(p^{\frac{3}{4}} \div p^{-\frac{2}{5}}\)                                                
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(p^{\frac{19}{20}}\)                                                                          
                 \end{solutionordottedlines}                                                                    
                 \part \(q^{\frac{3}{2}} \div q^{-\frac{2}{3}}\)                                                
                 \begin{solutionordottedlines}[2cm]                                                             
                     \(q^{\frac{13}{6}}\)                                                                           
                 \end{solutionordottedlines}                                                                    
             \end{multicols}                                                                                    
         \end{parts}                                                                                            
     \end{questions}                                                                                            
 \end{exercisebox}                                                                                              
