\begin{center}\begin{large}The beauty of powers\end{large}\end{center}

When you perform repeated multiplication with the same number, you can use power notation to summarise what you are doing. For example:

$$8 \times 8 \times 8 \times 8 = 8^4 \quad \text{(four } 8\text{s multiplied together)}$$

$$7 \times 7 \times 7 \times 7 \times 7 \times 7 = 7^6 \quad \text{(six } 7\text{s multiplied together)}$$

\begin{enumerate}
	\item For each of the following statements, tell whether it is True or False.
	
	\begin{enumerate}
		\item The number $8^{16}$ is $8$ times larger than the number $8^{15}$.
		\item The number $8^{10}$ is $10$ times larger than the number $8$.
	\end{enumerate}
	
	\item What is the value of the following expression?
	
	$$(-5)^{43} + (-1)^{43} + (5)^{43}$$

    \begin{enumerate}
    \begin{tabularx}{\linewidth}{*{4}X}
        \item $-1$ &
        \item $1$ &
        \item $0$ &
        \item $5$
    \end{tabularx}
    \end{enumerate}
\end{enumerate}

The first nine powers of $7$ are listed below. Notice how fast they grow!

The last digits of these numbers follow a rule or pattern. Study the pattern to answer the question.

\medskip

\begin{tabular}{r@{~$=$~}r}
	$7^1$ & $\textcolor{red}{7}$ \\
	$7^2$ & $4\textcolor{red}{9}$ \\
	$7^3$ & $34\textcolor{red}{3}$ \\
	$7^4$ & $2~40\textcolor{red}{1}$ \\
	$7^5$ & $16~80\textcolor{red}{7}$ \\
	$7^6$ & $117~64\textcolor{red}{9}$ \\
	$7^7$ & $823~54\textcolor{red}{3}$ \\
	$7^8$ & $5~764~80\textcolor{red}{1}$ \\
	$7^9$ & $40~353~60\textcolor{red}{7}$ \\
\end{tabular}

\begin{enumerate}
    \setcounter{enumi}{2}
	\item What is the last digit of the number $7^{190}$?

    \begin{enumerate}
    \begin{tabularx}{\linewidth}{*{4}X}
        \item $1$ &
        \item $3$ &
        \item $7$ &
        \item $9$
    \end{tabularx}
    \end{enumerate}
\end{enumerate}

\medskip
