1. Fill in the blanks.
a. (16)15 ÷ (16)19 = (1⁄16 ?)
\begin{align*} (16)^{15} \div (16)^{19} = \left(\frac{1}{16^?}\right) \end{align*}
Solution:
\begin{align*} &= (16)^{15} \div (16)^{19} \\ \\ &= (16)^{15-19} \\ \\ &= (16)^{-4} \\ \\ \text{Formula: } a ^{-m} & = \frac{1}{a^m} \\ \\ &= \left(\frac{1}{16^4}\right) \\ \\ (16)^{15} \div (16)^{19} & = \left(\frac{1}{16^\boxed4}\right) \end{align*}
b. (11⁄12)? ÷ (11⁄12)22 = 1⁄(11⁄12)2
\begin{align*} \left(\frac{11}{12}\right)^? \div \left(\frac{11}{12}\right)^{22} &= \frac{1}{\left(\frac{11}{12}\right)^2} \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } & \frac{1}{a^m} = a ^{-m} \\ \\ \left(\frac{11}{12}\right)^? \div \left(\frac{11}{12}\right)^{22} &= \left(\frac{11}{12}\right)^{-2} \\ \\ \left(\frac{11}{12}\right)^? &= \left(\frac{11}{12}\right)^{-2} \times \left(\frac{11}{12}\right)^{22} \\ \\ &= \left(\frac{11}{12}\right)^{-2 + 22} \\ \\ &= \left(\frac{11}{12}\right)^{20} \\ \\ \left(\frac{11}{12}\right)^\boxed{20} \div \left(\frac{11}{12}\right)^{22} &= \frac{1}{\left(\frac{11}{12}\right)^2} \\ \end{align*}
c. (1⁄32)? = (1⁄3)6
\begin{align*} \left(\frac{1}{3^2}\right)^? &= \left(\frac{1}{3}\right)^6 \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } & (a^m)^n = a ^{m \times n} \\ \\ &= \left(\frac{1}{3}\right)^6 \\ \\ &= \left(\frac{1}{3}\right)^{2 \times 3} \\ \\ &= \left(\frac{1}{3^2}\right)^{3} \\ \\ \left(\frac{1}{3^2}\right)^\boxed3 &= \left(\frac{1}{3}\right)^6 \\ \end{align*}
d. ((-1⁄2)4)2 = (-1⁄2)?
\begin{align*} \left(\left(-\frac{1}{2}\right)^4\right)^2 &= \left(-\frac{1}{2}\right)^{?} \\ \end{align*}
Solution:
\begin{align*} &= \left(\left(-\frac{1}{2}\right)^4\right)^2 \\ \\ \text{Formula: } (a^m)^n &= a ^{m \times n} \\ \\ &= \left(-\frac{1}{2}\right)^{4 \times 2} \\ \\ &= \left(-\frac{1}{2}\right)^8 \\ \\ \left(\left(-\frac{1}{2}\right)^4\right)^2 &= \left(-\frac{1}{2}\right)^\boxed{8} \\ \end{align*}
e. (1⁄12)5 ÷ (1⁄12)5 = (1⁄12)?
\begin{align*} \left(\frac{1}{12}\right)^5 \div \left(\frac{1}{12}\right)^5 & = \left(\frac{1}{12}\right)^? \\ \end{align*}
Solution:
\begin{align*} & = \left(\frac{1}{12}\right)^5 \div \left(\frac{1}{12}\right)^5 \\ \\ & \text{Formula:} \quad a^m \div a^n = a^{m-n} \\ \\ &= \left(\frac{1}{12}\right)^{5-5} \\ & = \left(\frac{1}{12}\right)^0 \\ \left(\frac{1}{12}\right)^5 \div \left(\frac{1}{12}\right)^5 & = \left(\frac{1}{12}\right)^\boxed{0} \\ \end{align*}
f. ((-1⁄7)0)7 = (-1⁄7)?
\begin{align*} \left(\left(-\frac{1}{7}\right)^0\right)^7 &= \left(-\frac{1}{7}\right)^{?} \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } (a^m)^n &= a ^{m \times n} \\ \\ &= \left(-\frac{1}{7}\right)^{0 \times 7} \\ \\ &= \left(-\frac{1}{7}\right)^0 \\ \\ \left(\left(-\frac{1}{7}\right)^0\right)^7 &= \left(-\frac{1}{7}\right)^\boxed{0} \\ \end{align*}
2. Simplify and express the result in exponential form.
a. ((9⁄2)0)5
\begin{align*} \left(\left(\frac{9}{2}\right)^0\right)^5 \end{align*}
Solution:
\begin{align*} \text{Formula: } (a^m)^n &= a ^{m \times n} \\ \\ & = \left(\left(\frac{9}{2}\right)^0\right)^5 \\ \\ &= \left(\frac{9}{2}\right)^{0 \times 5} \\ \\ &= \left(\frac{9}{2}\right)^{0} \\ \\ \left(\left(\frac{9}{2}\right)^0\right)^5 & = \left(\frac{9}{2}\right)^{0} \\ \\ \end{align*}
b. ((-5⁄17)6)3
\begin{align*} \left(\left(-\frac{5}{17}\right)^6\right)^3 \\ \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } (a^m)^n &= a^{m \times n} \\ \\ & = \left(\left(-\frac{5}{17}\right)^6\right)^3 \\ \\ &= \left(-\frac{5}{17}\right)^{6 \times 3} \\ \\ &= \left(-\frac{5}{17}\right)^{18} \\ \\ \left(\left(-\frac{5}{17}\right)^6\right)^3 & = \left(-\frac{5}{17}\right)^{18} \\ \\ \end{align*}
c. ((4⁄7)4)4
\begin{align*} \left(\left(\frac{4}{7}\right)^4\right)^4 \\ \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } (a^m)^n &= a^{m \times n} \\ \\ & = \left(\left(\frac{4}{7}\right)^4\right)^4 \\ \\ &= \left(\frac{4}{7}\right)^{4 \times 4} \\ \\ &= \left(\frac{4}{7}\right)^{16} \\ \\ \left(\left(\frac{4}{7}\right)^4\right)^4 & = \left(\frac{4}{7}\right)^{16} \\ \\ \end{align*}
d. [(-2⁄5)3 x (-2⁄5)2 ]4
\begin{align*} \left[ \left(-\frac{2}{5}\right)^3 \times \left(-\frac{2}{5}\right)^2 \right]^4 \\ \\ \end{align*}
Solution:
\begin{align*} & = \left[ \left(-\frac{2}{5}\right)^3 \times \left(-\frac{2}{5}\right)^2 \right]^4 \\ \\ & = \left[ \left(-\frac{2}{5}\right)^{3 + 2} \right]^4 \\ \\ & = \left[ \left(-\frac{2}{5}\right)^5 \right]^4 \\ \\ & = \left(-\frac{2}{5}\right)^{5 \times 4} \\ \\ \left[ \left(-\frac{2}{5}\right)^3 \times \left(-\frac{2}{5}\right)^2 \right]^4 & = \left(-\frac{2}{5}\right)^{20} \\ \end{align*}
e. (3⁄5)3 ÷ (3⁄5)8
\begin{align*} \left(\frac{3}{5}\right)^3 \div \left(\frac{3}{5}\right)^8 \\ \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } a^m \div a^n &= \frac{1}{a^{n- m}} \\ \\ & = \left(\frac{3}{5}\right)^3 \div \left(\frac{3}{5}\right)^8 \\ \\ & = \frac{1}{\left(\frac{3}{5}\right)^{8 - 3}} \\ \\ & = \frac{1}{\left(\frac{3}{5}\right)^{5}} \\ \\ & = \left(\frac{5}{3}\right)^{5} \\ \\ \left(\frac{3}{5}\right)^3 \div \left(\frac{3}{5}\right)^8 & = \left(\frac{5}{3}\right)^{5} \end{align*}
f. [(12⁄5)0 x (12⁄5)]5
\begin{align*} \left[\left(\frac{12}{5}\right)^0 \times \left(\frac{12}{5}\right)\right]^5 \\ \\ \end{align*}
Solution:
\begin{align*} & = \left[\left(\frac{12}{5}\right)^{0 + 1}\right]^5 \\ \\ & = \left[\left(\frac{12}{5}\right)^{1}\right]^5 \\ \\ & = \left(\frac{12}{5}\right)^{1 \times 5} \\ \\ & = \left(\frac{12}{5}\right)^{5} \\ \\ \left[\left(\frac{12}{5}\right)^0 \times \left(\frac{12}{5}\right)\right]^5 & = \left(\frac{12}{5}\right)^{5} \\ \\ \end{align*}
3. Evaluate
a. (-3⁄4)3 ÷ (-3⁄4)5
\begin{align*} \left(-\frac{3}{4}\right)^3 \div \left(-\frac{3}{4}\right)^5 \\ \\ \end{align*}
Solution:
\begin{align*} \text{Formula: } a^m \div a^n &= \frac{1}{a^{n- m}} \\ \\ &= \left(-\frac{3}{4}\right)^3 \div \left(-\frac{3}{4}\right)^5 \\ \\ & = \frac{1}{\left(-\frac{3}{4}\right)^{5 - 3}} \\ \\ & = \frac{1}{\left(-\frac{3}{4}\right)^{2}} \\ \\ & = {\left(-\frac{4}{3}\right)^{2}} \\ \\ & = \left(-\frac{4}{3}\right) \times \left(-\frac{4}{3}\right) \\ \\ \left(-\frac{3}{4}\right)^3 \div \left(-\frac{3}{4}\right)^5 & = \left(\frac{16}{9}\right) \\ \\ \end{align*}
b. (1⁄52)2 x (1⁄5)
\begin{align*} \left(\frac{1}{5^2}\right)^2 \times \frac{1}{5} \\ \\ \end{align*}
Solution:
\begin{align*} & = \left(\frac{1}{5^2}\right) \times \left(\frac{1}{5^2}\right) \times \frac{1}{5} \\ \\ & = \left(\frac{1}{25}\right) \times \left(\frac{1}{25}\right) \times \frac{1}{5} \\ \\ & = \frac{1}{625} \times \frac{1}{5} \\ \\ & = \frac{1}{3125} \\ \\ \left(\frac{1}{5^2}\right)^2 \times \frac{1}{5} & = \frac{1}{3125} \\ \\ \end{align*}
c. [(-5⁄6)2]2 ÷ (-5⁄6)2
\begin{align*} \left[\left(-\frac{5}{6}\right)^2\right]^2 \div \left(-\frac{5}{6}\right)^2 \end{align*}
Solution:
\begin{align*} &= \left(-\frac{5}{6}\right)^{4} \div \left(-\frac{5}{6}\right)^2 \\ \\ &= \left(-\frac{5}{6}\right)^{4-2} \\ \\ &= \left(-\frac{5}{6}\right)^{2} \\ \\ &= \left(-\frac{5}{6}\right) \times \left(-\frac{5}{6}\right) \\ \\ &= \frac{25}{36} \\ \left[\left(-\frac{5}{6}\right)^2\right]^2 \div \left(-\frac{5}{6}\right)^2 &= \frac{25}{36} \end{align*}
d. (2/3)2 ÷ [(2/3)2]0
\begin{align*} \left(\frac{2}{3}\right)^2 \div \left[\left(\frac{2}{3}\right)^2\right]^0 \end{align*}
Solution:
\begin{align*} &= \left(\frac{2}{3}\right)^2 \div \left(\frac{2}{3}\right)^{2 \times 0 } \\ \\ &= \left(\frac{2}{3}\right)^2 \div \left(\frac{2}{3}\right)^{0} \\ \\ &= \left(\frac{2}{3}\right)^{2 - 0} \\ \\ &= \left(\frac{2}{3}\right)^{2} \\ \\ &= \left(\frac{2}{3}\right) \times \left(\frac{2}{3}\right) \\ \\ &= \left(\frac{4}{9}\right) \\ \\ \left(\frac{2}{3}\right)^2 \div \left[\left(\frac{2}{3}\right)^2\right]^0 &= \left(\frac{4}{9}\right) \\ \\ \end{align*}
e.(1/2)5⁄(1/2)3 - (1/2)6⁄(1/2)5
\begin{align*} \frac{\left(\frac{1}{2}\right)^5}{\left(\frac{1}{2}\right)^3} - \frac{\left(\frac{1}{2}\right)^6}{\left(\frac{1}{2}\right)^5} \\ \end{align*}
Solution:
\begin{align*} &= \left(\frac{1}{2}\right)^{5-3} - \left(\frac{1}{2}\right)^{6-5} \\ \\ &= \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^1 \\ \\ &= \frac{1}{4} - \frac{1}{2} \\ \\ &= \frac{1 - 2}{4}\\ \\ &= -\frac{1}{4} \\ \\ \frac{\left(\frac{1}{2}\right)^5}{\left(\frac{1}{2}\right)^3} - \frac{\left(\frac{1}{2}\right)^6}{\left(\frac{1}{2}\right)^5} &= -\frac{1}{4} \\ \\ \end{align*}
4. Find the value of:
a. (5⁄7)3 x 2 - 6
\begin{align*} \left(\frac{5}{7}\right)^{3 \times 2 - 6} \end{align*}
Solution:
\begin{align*} & = \left(\frac{5}{7}\right)^{6 - 6} \\ \\ & = \left(\frac{5}{7}\right)^{0} \\ \\ & = 1 \\ \\ \left(\frac{5}{7}\right)^{3 \times 2 - 6} & = 1 \end{align*}
b. (30 + 40) x 50
\begin{align*} (3^0 + 4^0) \times 5^0 \\ \end{align*}
Solution:
\begin{align*} &= (3^0 + 4^0) \times 5^0 \\ &= (1 + 1) \times 1 \\ &= 2 \times 1 \\ &= 2 \\ \end{align*}
c. 10 x 20 + 30 x 40 + 50 x 60
\begin{align*} 1^0 \times 2^0 + 3^0 \times 4^0 + 5^0 \times 6^0 \\ \end{align*}
Solution:
\begin{align*} &= 1^0 \times 2^0 + 3^0 \times 4^0 + 5^0 \times 6^0 \\ &= (1) \times (1) + (1) \times (1) + (1) \times (1) \\ &= 1 + 1 + 1 \\ &= 3 \\ \end{align*}
d. (-5⁄9)9 - 3 x 2 - 3
\begin{align*} \left(-\frac{5}{9}\right)^{9 - 3 \times 2 - 3} \\ \end{align*}
Solution:
\begin{align*} &= \left(-\frac{5}{9}\right)^{9 - 3 \times 2 - 3} \\ \\ &= \left(-\frac{5}{9}\right)^{9 - 6 - 3} \\ \\ &= \left(-\frac{5}{9}\right)^{3 - 3} \\ \\ &= \left(-\frac{5}{9}\right)^{0} \\ \\ &= 1 \end{align*}
5. Find the value of x so that -
a. (3/4)2x + 1 = [(3/4)3]3
\begin{align*} \left(\frac{3}{4}\right)^{2x + 1} &= \left[\left(\frac{3}{4}\right)^3\right]^3 \\ \end{align*}
Solution:
\begin{align*} \left(\frac{3}{4}\right)^{2x + 1} &= \left(\frac{3}{4}\right)^{3 \times 3} \\ \\ \left(\frac{3}{4}\right)^{2x + 1} &= \left(\frac{3}{4}\right)^{9} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 2x + 1 &= 9 \\ \\ 2x &= 9 - 1 \\ \\ 2x &= 8 \\ \\ x &= \frac{8}{2} \\ \\ x &= 4 \end{align*}
b. (2/5)3 x (2/5)6 = (2/5)3x
\begin{align*} \left(\frac{2}{5}\right)^{3} \times \left(\frac{2}{5}\right)^{6} &= \left(\frac{2}{5}\right)^{3x} \\ \end{align*}
Solution:
\begin{align*} \left(\frac{2}{5}\right)^{3 + 6} &= \left(\frac{2}{5}\right)^{3x} \\ \\ \left(\frac{2}{5}\right)^{9} &= \left(\frac{2}{5}\right)^{3x} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 9 &= 3x \\ x &= \frac{9}{3} \\ x &= 3 \end{align*}
c. (-1/5)20 ÷ (-1/5)15 = (-1/5)5x
\begin{align*} \left(-\frac{1}{5}\right)^{20} \div {\left(-\frac{1}{5}\right)^{15}} &= \left(-\frac{1}{5}\right)^{5x} \\ \end{align*}
Solution:
\begin{align*} \left(-\frac{1}{5}\right)^{20 - 15} &= \left(-\frac{1}{5}\right)^{5x} \\ \\ \left(-\frac{1}{5}\right)^{5} &= \left(-\frac{1}{5}\right)^{5x} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 5 &= 5x \\ x &= 1 \end{align*}
d. (1/16) x (1/2)2 = (1/2)3(x - 2)
\begin{align*} \left(\frac{1}{16}\right) \times \left(\frac{1}{2}\right)^2 &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \end{align*}
Solution:
\begin{align*} \left(\frac{1}{16}\right) \times \left(\frac{1}{2}\right)^2 &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \\ \left(\frac{1}{2^4}\right) \times \left(\frac{1}{2}\right)^2 &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \\ \left(\frac{1}{2}\right)^4 \times \left(\frac{1}{2}\right)^2 &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \\ \left(\frac{1}{2}\right)^{4+2} &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \\ \left(\frac{1}{2}\right)^6 &= \left(\frac{1}{2}\right)^{3(x - 2)} \\ \\ \text{Since bases are same, } & \text{their exponents must be equal.} \\ \\ 3(x - 2) & = 6 \\ x - 2 & = \frac{6}{3} \\ x - 2 & = 2 \\ x & = 2 + 2 \\ x & = 4 \end{align*}
6. Which of the following statements are true?
a. (0.6)8 ÷ (0.6)7 = (0.6)2
\begin{align*} \left(0.6\right)^8 \div \left(0.6\right)^7 & = \left(0.6\right)^2 \end{align*}
Solution: False
\begin{align*} & = \left(0.6\right)^8 \div \left(0.6\right)^7 \\ \\ \text{Formula} \\ a^m \div a^n &= a^{m-n} \\ \\ &= \left(0.6\right)^{8-7} \\ \\ \left(0.6\right)^8 \div \left(0.6\right)^7 &= \left(0.6\right)^1 \\ \end{align*}
b. (12 / 13)6 ÷ (12 / 13)3 = (12 / 13)3
\begin{align*} \left(\frac{12}{13}\right)^6 \div \left(\frac{12}{13}\right)^3 &= \left(\frac{12}{13}\right)^3 \\ \end{align*}
Solution: True
\begin{align*} & = \left(\frac{12}{13}\right)^6 \div \left(\frac{12}{13}\right)^3 \\ \\ \text{Formula } \\ a^m \div a^n &= a^{m-n} \\ \\ &= \left(\frac{12}{13}\right)^{6-3} \\ \\ \left(\frac{12}{13}\right)^6 \div \left(\frac{12}{13}\right)^3 &= \left(\frac{12}{13}\right)^3 \\ \end{align*}
c. The reciprocal of (7 / 5)12 is (5 / 7)12
\begin{align*} \text{Reciprocal of }\left(\frac{7}{5}\right)^{12} & \text{ is } \left(\frac{5}{7}\right)^{12} \\ \end{align*}
Solution: True
d. (5 + 5)5 = 55 + 55
\begin{align*} (5 + 5)^5 = 5^5 + 5^5 \end{align*}
Solution: False
\begin{align*} \text{LHS} \\ & = (5 + 5)^5 \\ &= 10^5 \\ &= 100000 \\ \\ \text{RHS} \\ &= 5^5 + 5^5 \\ &= 3125 + 3125 \\ &= 6250 \\ \\ (5 + 5)^5 &\neq 5^5 + 5^5 \end{align*}
e. [(1/4)4 ÷ (1/4)3] ÷ (1/4) = 1/4
\begin{align*} \left[\left(\frac{1}{4}\right)^4 \div \left(\frac{1}{4}\right)^3\right] \div \left(\frac{1}{4}\right) &= \frac{1}{4} \\ \\ \end{align*}
Solution: False
\begin{align*} \text{Formula } \\ a^m \div a^n &= a^{m-n} \\ \\ \text{LHS} \\ & = \left[\left(\frac{1}{4}\right)^4 \div \left(\frac{1}{4}\right)^3\right] \div \left(\frac{1}{4}\right) \\ \\ & = \left[\left(\frac{1}{4}\right)^{4-3}\right] \div \left(\frac{1}{4}\right) \\ \\ & = \left[\left(\frac{1}{4}\right)^1\right] \div \left(\frac{1}{4}\right) \\ \\ & = \left(\frac{1}{4}\right)^{1-1} \\ \\ & = \left(\frac{1}{4}\right)^0 \\ \\ & = 1 \\ \text{RHS} &= \frac{1}{4} \\ \\ \left[\left(\frac{1}{4}\right)^4 \div \left(\frac{1}{4}\right)^3\right] \div \left(\frac{1}{4}\right) &\neq \frac{1}{4} \\ \\ \end{align*}
f. (1/7 x 1/72) ÷ 1/73 = 1
\begin{align*} \left(\frac{1}{7} \times \frac{1}{7^2}\right) \div \left(\frac{1}{7^3}\right) &= 1 \\ \\ \end{align*}
Solution: True
\begin{align*} & = \left(\frac{1}{7} \times \frac{1}{49}\right) \div \frac{1}{343} \\ \\ & = \frac{1}{343} \div \frac{1}{343} \\ \\ & = \frac{1}{343} \times 343 \\ \\ & = 1 \\ \left(\frac{1}{7} \times \frac{1}{7^2}\right) \div \left(\frac{1}{7^3}\right) &= 1 \\ \end{align*}