1. Write the base and exponent in each of the following.
a. (-1⁄3)3
\begin{align*} \left(-\frac{1}{3}\right)^3 \\ \end{align*}
\begin{align*} \text{Base = } & -\frac{1}{3} \\ \\ \text{Exponent = } & \quad 3 \end{align*}
b. (-4⁄7)6
\begin{align*} \left(-\frac{4}{7}\right)^6 \\ \end{align*}
\begin{align*} \text{Base = } & -\frac{4}{7} \\ \\ \text{Exponent = } & \quad 6 \end{align*}
c. (2⁄9)5
\begin{align*} \left(\frac{2}{9}\right)^5 \\ \end{align*}
\begin{align*} \text{Base = } & \frac{2}{9} \\ \\ \text{Exponent = } & 5 \end{align*}
d. (15⁄19)3
\begin{align*} \left(\frac{15}{19}\right)^3 \\ \end{align*}
\begin{align*} \text{Base = } & \frac{15}{19} \\ \\ \text{Exponent = } & 3 \end{align*}
e. (-15)4
\begin{align*} (-15)^4 \\ \end{align*}
\begin{align*} \text{Base = } & -15 \\ \\ \text{Exponent = } & \quad 4 \end{align*}
f. (-2⁄3)
\begin{align*} \left(-\frac{2}{3}\right) \\ \end{align*}
\begin{align*} \text{Base = } & \left(-\frac{2}{3}\right) \\ \\ \text{Exponent = } & \quad 1 \end{align*}
2. Express the following in exponential form.
a. 5⁄6 x 5⁄6
\begin{align*} \frac{5}{6} \times \frac{5}{6} & = \left(\frac{5}{6}\right)^2 \\ \end{align*}
b. 9⁄2 x 9⁄2 x 9⁄2 x 9⁄2
\begin{align*} \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} \times \frac{9}{2} & = \left(\frac{9}{2}\right)^4 \\ \end{align*}
c. (-7⁄8) x (-7⁄8) x (-7⁄8)
\begin{align*} \left(-\frac{7}{8}\right) \times \left(-\frac{7}{8}\right) \times \left(-\frac{7}{8}\right) & = \left(-\frac{7}{8}\right)^3 \\ \end{align*}
d. (-1⁄2) x (-1⁄2) x (-1⁄2) x (-1⁄2) x (-1⁄2)
\begin{align*} \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) & = \left(-\frac{1}{2}\right)^5 \\ \end{align*}
e. 1.8 x 1.8 x 1.8 x 1.8 x 1.8 x 1.8 x 1.8
\begin{align*} 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 \times 1.8 & = (1.8)^7 \\ \end{align*}
f. 11⁄12 x 11⁄12 x 11⁄12 x 11⁄12 x 11⁄12 x 11⁄12
\begin{align*} \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} \times \frac{11}{12} & = \left(\frac{11}{12}\right)^6 \\ \end{align*}
3. Express the following as rational numbers in the form p⁄q
a. (5⁄6)3
\begin{align*} \left(\frac{5}{6}\right)^3 \\ \end{align*}
\begin{align*} & = \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6} \\ \\ & = \frac{125}{216} \end{align*}
b. (-12⁄13)2
\begin{align*} \left(-\frac{12}{13}\right)^2 \\ \end{align*}
\begin{align*} & = \left(-\frac{12}{13}\right) \times \left(-\frac{12}{13}\right) \\ \\ & = \frac{144}{169} \end{align*}
c. (4⁄9)4
\begin{align*} \left(\frac{4}{9}\right)^4 \\ \end{align*}
\begin{align*} & = \left(\frac{4}{9}\right) \times \left(\frac{4}{9}\right) \times \left(\frac{4}{9}\right) \times \left(\frac{4}{9}\right) \\ \\ & = \frac{256}{6561} \end{align*}
d. (-1⁄2)5
\begin{align*} \left(-\frac{1}{2}\right)^5 \\ \end{align*}
\begin{align*} & = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \\ \\ & = \left(-\frac{1}{32}\right) \\ \end{align*}
e. (1⁄4)4
\begin{align*} \left(\frac{1}{4}\right)^4 \\ \end{align*}
\begin{align*} & = \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \times \left(\frac{1}{4}\right) \\ \\ & = \left(\frac{1}{256}\right) \\ \end{align*}
f. (3⁄5)3
\begin{align*} \left(\frac{3}{5}\right)^3 \\ \end{align*}
\begin{align*} & = \left(\frac{3}{5}\right) \times \left(\frac{3}{5}\right) \times \left(\frac{3}{5}\right) \\ \\ & = \left(\frac{27}{125}\right) \\ \end{align*}
4. Express the following as powers of rational numbers.
a. (81⁄625)
\begin{align*} \left(\frac{81}{625}\right) \\ \end{align*}
\begin{align*} \text{Factorisation of } 81 \text{ and } 625 \\ \end{align*} \begin{align*} \begin{array}{r|l} 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 5 & 625 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{81}{625} &= \frac{3 \times 3 \times 3 \times 3}{5 \times 5 \times 5 \times 5} \\ \\ &= \frac{3^4}{5^4} \\ \\ \frac{81}{625} &= \left(\frac{3}{5}\right)^4 \end{align*}
b. (-8⁄125)
\begin{align*} \left(-\frac{8}{125}\right) \\ \end{align*}
\begin{align*} \text{Factorisation of } -8 \text{ and } 125 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} -\frac{8}{125} &= -\frac{2 \times 2 \times 2}{5 \times 5 \times 5} \\ \\ &= -\frac{2^3}{5^3} \\ \\ -\frac{8}{125} &= \left(-\frac{2}{5}\right)^3 \end{align*}
c. (-343⁄512)
\begin{align*} \left(-\frac{343}{512}\right) \\ \end{align*}
\begin{align*} \text{Factorisation of } -343 \text{ and } 512 \\ \end{align*} \begin{align*} \begin{array}{r|l} 7 & 343 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & -1 \\ \end{array} && \begin{array}{r|l} 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \left(-\frac{343}{512}\right) &= -\frac{7 \times 7 \times 7}{\boxed{2 \times 2 \times 2} \times \boxed{2 \times 2 \times 2} \times \boxed{2 \times 2 \times 2}} \\ \\ &= -\frac{7^3}{8 \times 8 \times 8} \\ \\ &= -\frac{7^3}{8^3} \\ \\ \left(-\frac{343}{512}\right) &= \left(-\frac{7}{8}\right)^3 \end{align*}
d. (32⁄243)
\begin{align*} \left(\frac{32}{243}\right) \\ \end{align*}
\begin{align*} \text{Prime factorisation of } 32 \text{ and } 243 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{32}{243} &= \frac{2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3} \\ \\ &= \frac{2^5}{3^5} \\ \\ \frac{32}{243} &= \left(\frac{2}{3}\right)^5 \end{align*}
e. (-1⁄216)
\begin{align*} \left(-\frac{1}{216}\right) \\ \end{align*}
\begin{align*} \text{Prime factorisation of } 216 \\ \end{align*} \begin{align*} \begin{array}{r|l} 2 & 216 \\ \hline 2 & 108 \\ \hline 2 & 54 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} -\frac{1}{216} &= -\frac{1^3}{2 \times 2 \times 2 \times 3 \times 3 \times 3} \\ \\ &= -\frac{1^3}{\boxed{2 \times 3} \times \boxed{2 \times 3} \times \boxed{2 \times 3}} \\ \\ &= \left(-\frac{1^3}{6^3}\right) \\ \\ -\frac{1}{216} &= -\left(\frac{1}{6}\right)^3 \end{align*}
f. (729⁄1000)
\begin{align*} \left(\frac{729}{1000}\right) \\ \end{align*}
\begin{align*} \text{Prime factorisation of } 729 \text{ and } 1000 \\ \end{align*} \begin{align*} \begin{array}{r|l} 3 & 729 \\ \hline 3 & 243 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} && \begin{array}{r|l} 2 & 1000 \\ \hline 2 & 500 \\ \hline 2 & 250 \\ \hline 2 & 125 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \end{align*} \begin{align*} \frac{729}{1000} &= \frac{3 \times 3 \times 3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 5 \times 5 \times 5} \\ \\ &= \frac{\boxed{3 \times 3} \times \boxed{3 \times 3} \times \boxed{3 \times 3}}{\boxed{2 \times 5} \times \boxed{2 \times 5} \times \boxed{2 \times 5}} \\ \\ &= \frac{9^3}{10^3} \\ \\ \frac{729}{1000} &= \left(\frac{9}{10}\right)^3 \end{align*}