1. Which of the following statements are true?
a. (+3⁄4)3 x (-3⁄4)3 = (-3⁄4)6
\begin{align*} \left(+\frac{3}{4}\right)^3 \times \left(-\frac{3}{4}\right)^3 & = \left(-\frac{3}{4}\right)^6 \\ \end{align*}
Solution: False
Base is not the same.
b. (4⁄7)5 x (4⁄7)3 = (4⁄7)8
\begin{align*} \left(\frac{4}{7}\right)^5 \times \left(\frac{4}{7}\right)^3 = \left(\frac{4}{7}\right)^8 \\ \end{align*}
Solution : True
c. (-1⁄2)4 ÷ (-1⁄2)3 = (-1⁄2)
\begin{align*} \left(-\frac{1}{2}\right)^4 \div \left(-\frac{1}{2}\right)^3 = \left(-\frac{1}{2}\right) \end{align*}
Solution : True
d. (6⁄7)6 ÷ (6⁄7)0 = (6⁄7)0
\begin{align*} \left(\frac{6}{7}\right)^6 \div \left(\frac{6}{7}\right)^0 = \left(\frac{6}{7}\right)^0 \end{align*}
Solution : False
\begin{align*} \left(\frac{6}{7}\right)^6 \div \left(\frac{6}{7}\right)^0 = \left(\frac{6}{7}\right)^\boxed{6} \end{align*}
2. Fill in the blank in each of the following so as to make the statement true.
a. (-2⁄3) x (-2⁄3)3 = (-2⁄3)?
\begin{align*} \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)^3 & = \left(-\frac{2}{3}\right)^\boxed{?} \end{align*}
\begin{align*} & = \left(-\frac{2}{3}\right)^1 \times \left(-\frac{2}{3}\right)^3 \\ \\ & = \left(-\frac{2}{3}\right)^\boxed{1 + 3} \\ \\ & = \left(-\frac{2}{3}\right)^\boxed{4} \\ \\ \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)^3 & = \left(-\frac{2}{3}\right)^\boxed{4} \end{align*}
b. (4⁄3)? ÷ (4⁄5)9 = (4⁄5)11
\begin{align*} \left(\frac{4}{5}\right)^\boxed{?} \times \left(\frac{4}{5}\right)^9 & = \left(\frac{4}{5}\right)^{11} \end{align*}
\begin{align*} \left(\frac{4}{5}\right)^\boxed{?} & = \left(\frac{4}{5}\right)^{11} \div \left(\frac{4}{5}\right)^9 \\ \\ & = \left(\frac{4}{5}\right)^\boxed{11 - 9} \\ \\ & = \left(\frac{4}{5}\right)^2 \\ \\ \left(\frac{4}{5}\right)^\boxed{2} \times \left(\frac{4}{5}\right)^9 & = \left(\frac{4}{5}\right)^{11} \end{align*}
c. (-3⁄7)15 ÷ (-3⁄7)? = (-3⁄7)1
\begin{align*} \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^\boxed{?} & = \left(-\frac{3}{7}\right) \end{align*}
\begin{align*} \left(-\frac{3}{7}\right)^\boxed{?} & = \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^1 \\ \\ & = \left(-\frac{3}{7}\right)^\boxed{15 - 1} \\ \\ & = \left(-\frac{3}{7}\right)^{14} \\ \\ \left(-\frac{3}{7}\right)^{15} \div \left(-\frac{3}{7}\right)^\boxed{14} & = \left(-\frac{3}{7}\right) \end{align*}
d. (-1⁄10)? ÷ (-1⁄10)8 = (-1⁄10)8
\begin{align*} \left(-\frac{1}{10}\right)^\boxed{?} \div \left(-\frac{1}{10}\right)^8 &= \left(-\frac{1}{10}\right)^8 \\ \end{align*}
\begin{align*} \left(-\frac{1}{10}\right)^\boxed{?} &= \left(-\frac{1}{10}\right)^8 \times \left(-\frac{1}{10}\right)^8 \\ \\ &= \left(-\frac{1}{10}\right)^\boxed{8+8} \\ \\ &= \left(-\frac{1}{10}\right)^{16} \\ \\ \left(-\frac{1}{10}\right)^\boxed{16} \div \left(-\frac{1}{10}\right)^8 &= \left(-\frac{1}{10}\right)^8 \\ \end{align*}
e. (2⁄9)6 ÷ (2⁄9)0 = (2⁄9)?
\begin{align*} \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 &= \left(\frac{2}{9}\right)^\boxed{?} \\ \end{align*}
\begin{align*} \left(\frac{2}{9}\right)^\boxed{?} & = \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 \\ \\ & = \left(\frac{2}{9}\right)^\boxed{6 - 0} \\ \\ & = \left(\frac{2}{9}\right)^\boxed{6} \\ \\ \left(\frac{2}{9}\right)^6 \div \left(\frac{2}{9}\right)^0 &= \left(\frac{2}{9}\right)^\boxed{6} \\ \end{align*}
f. (-12⁄13)2 x (-12⁄13)? = (-12⁄13)5
\begin{align*} \left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^\boxed{?} &= \left(-\frac{12}{13}\right)^5 \\ \end{align*}
\begin{align*} \left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^\boxed{?} &= \left(-\frac{12}{13}\right)^5 \\ \\ &= \left(-\frac{12}{13}\right)^5 \div \left(-\frac{12}{13}\right)^2 \\ \\ &= \left(-\frac{12}{13}\right)^\boxed{5 - 2} \\ \\ &= \left(-\frac{12}{13}\right)^\boxed{3} \\ \\ \left(-\frac{12}{13}\right)^2 \times \left(-\frac{12}{13}\right)^\boxed{3} &= \left(-\frac{12}{13}\right)^5 \\ \end{align*}
3. Simplify and express the result in exponential form.
a. (-3⁄7)3 x (-3⁄7)4
\begin{align*} \left(-\frac{3}{7}\right)^3 \times \left(-\frac{3}{7}\right)^4 \end{align*}
\begin{align*} &= \left(-\frac{3}{7}\right)^{3 + 4} \\ \\ &= \left(-\frac{3}{7}\right)^7 \\ \end{align*}
b. (11⁄12)15 x (11⁄12)5 x (11⁄12)10
\begin{align*} \left(\frac{11}{12}\right)^{15} \times \left(\frac{11}{12}\right)^5 \times \left(\frac{11}{12}\right)^{10} \end{align*}
\begin{align*} &= \left(\frac{11}{12}\right)^{15 + 5 + 10} \\ &= \left(\frac{11}{12}\right)^{30} \end{align*}
c. (-9⁄11)9 ÷ (-9⁄11)7
\begin{align*} \left(-\frac{9}{11}\right)^9 \div \left(-\frac{9}{11}\right)^7 \end{align*}
\begin{align*} &= \left(-\frac{9}{11}\right)^{9 - 7} \\ \\ &= \left(-\frac{9}{11}\right)^{2} \end{align*}
d. (1⁄4)8 ÷ (1⁄4)6
\begin{align*} \left(\frac{1}{4}\right)^8 \div \left(\frac{1}{4}\right)^6 \end{align*}
\begin{align*} &= \left(\frac{1}{4}\right)^{8 - 6} \\ &= \left(\frac{1}{4}\right)^2 \end{align*}
4. Simplify and express the result as a rational number.
a. (5⁄7)4 ÷ (5⁄7)2
\begin{align*} \left(\frac{5}{7}\right)^4 \div \left(\frac{5}{7}\right)^2 \end{align*}
\begin{align*} &= \left(\frac{5}{7}\right)^{4 - 2} \\ \\ &= \left(\frac{5}{7}\right)^2 \\ \\ &= \left(\frac{5}{7}\right) \times \left(\frac{5}{7}\right) \\ \\ &= \frac{25}{49} \end{align*}
b. (-2⁄3)2 x (-2⁄3)3
\begin{align*} \left(-\frac{2}{3}\right)^2 \times \left(-\frac{2}{3}\right)^3 \end{align*}
\begin{align*} &= \left(-\frac{2}{3}\right)^{2+3} \\ \\ &= \left(-\frac{2}{3}\right)^5 \\ \\ &= \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \\ \\ &= -\frac{32}{243} \end{align*}
c. (3⁄4)3 x (3⁄4)2
\begin{align*} \left(\frac{3}{4}\right)^3 \times \left(\frac{3}{4}\right)^2 \end{align*}
\begin{align*} &= \left(\frac{3}{4}\right)^{3+2} \\ \\ &= \left(\frac{3}{4}\right)^5 \\ \\ &= \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) \\ \\ &= \frac{243}{1024} \end{align*}
d. (-3⁄5)6 ÷ (-3⁄5)3
\begin{align*} \left(-\frac{3}{5}\right)^6 \div \left(-\frac{3}{5}\right)^3 \end{align*}
\begin{align*} &= \left(-\frac{3}{5}\right)^{6-3} \\ \\ &= \left(-\frac{3}{5}\right)^3 \\ \\ &= \left(-\frac{3}{5}\right) \times \left(-\frac{3}{5}\right) \times \left(-\frac{3}{5}\right) \\ \\ &= -\frac{27}{125} \end{align*}
e. (-1⁄10)4 x (-1⁄10)2
\begin{align*} \left(-\frac{1}{10}\right)^4 \times \left(-\frac{1}{10}\right)^2 \end{align*}
\begin{align*} &= \left(-\frac{1}{10}\right)^{4 + 2} \\ \\ &= \left(-\frac{1}{10}\right)^6 \\ \\ &= \left(-\frac{1}{10}\right) \times \left(-\frac{1}{10}\right) \times \left(-\frac{1}{10}\right) \times \left(-\frac{1}{10}\right) \times \left(-\frac{1}{10}\right) \times \left(-\frac{1}{10}\right) \\ \\ &= \frac{1}{1000000} \end{align*}
5. Evaluate
a. (3⁄4)3 x (2⁄3)2
\begin{align*} \left(\frac{3}{4}\right)^3 \times \left(\frac{2}{3}\right)^2 \end{align*}
\begin{align*} &= \frac{3}{4} \times \frac{\cancel3}{\cancelto{2}4} \times \frac{\cancel{3}}{\cancelto{2}{4}} \times \frac{\cancelto{1}2}{\cancel{3}} \times \frac{\cancelto{1}2}{\cancel3} \\ \\ &= \frac{3}{4 \times 2 \times 2} \\ \\ &= \frac{3}{16} \end{align*}
b. [(1⁄3)6 ÷ (1⁄3)5] ÷ (1⁄3)
\begin{align*} \left[\left(\frac{1}{3}\right)^6 \div \left(\frac{1}{3}\right)^5\right] \div \left(\frac{1}{3}\right) \\ \end{align*}
\begin{align*} &= \left[\left(\frac{1}{3}\right)^{6-5}\right] \div \left(\frac{1}{3}\right) \\ \\ &= \left(\frac{1}{3}\right)^{1} \div \left(\frac{1}{3}\right) \\ \\ &= \left(\frac{1}{3}\right)^{1-1} \\ \\ &= \left(\frac{1}{3}\right)^0 \\ \\ &= 1 \end{align*}
c. (24 x 25) ÷ 28
\begin{align*} ({2^4 \times 2^5}) \div {2^8} \\ \end{align*}
\begin{align*} &= ({2^{4+5}}) \div {2^8} \\ \\ &= {2^{9}} \div {2^8} \\ \\ &= {2^{9-8}} \\ \\ &= 2^{1} \\ \\ &= 2 \end{align*}
d. (42 - 32) ÷ (1⁄7)2
\begin{align*} ({4^2 - 3^2}) \div {\left(\frac{1}{7}\right)^2} \\ \end{align*}
\begin{align*} &= ({16 - 9}) \div {\left(\frac{1}{7}\right)} \times {\left(\frac{1}{7}\right)} \\ \\ &= {7} \div {\frac{1}{49}} \\ \\ &= 7 \times 49 \\ \\ &= 343 \end{align*}