1. **State the problem:** Simplify and compute the given expressions step-by-step. 2. **Simplify each expression:** a) $5^{22}$ remains as is. b) $\dfrac{2^{23}}{4^3} = \dfrac{2^{23}}{(2^2)^3} = \dfrac{2^{23}}{2^{6}} = 2^{17}$. c) $\dfrac{125^3}{25} = \dfrac{(5^3)^3}{5^2} = \dfrac{5^9}{5^2} = 5^{7}$. d) $\dfrac{27^3 \cdot 3^2}{9^3} = \dfrac{(3^3)^3 \cdot 3^2}{(3^2)^3} = \dfrac{3^{9} \cdot 3^{2}}{3^{6}} = 3^{9+2-6} = 3^{5}$. e) $\dfrac{125^2 \cdot 25^3}{5^4} = \dfrac{(5^3)^2 \cdot (5^2)^3}{5^4} = \dfrac{5^{6} \cdot 5^{6}}{5^{4}} = 5^{6+6-4} = 5^{8}$. f) $\dfrac{(\frac{1}{8})^{3} \cdot 64^4}{4^3} = \dfrac{(2^{-3})^{3} \cdot (2^{6})^{4}}{2^{6}} = \dfrac{2^{-9} \cdot 2^{24}}{2^{6}} = 2^{-9 +24 - 6} = 2^{9}$. 3. **Solve Bài 19:** Simplify each expression. a) $(\frac{2}{3})^3 \cdot (\frac{8}{27})^3 = (\frac{2}{3} \cdot \frac{8}{27})^3 = (\frac{16}{81})^3 = \frac{16^3}{81^3}$. b) $\dfrac{(\frac{7}{-5})^5}{(-\frac{14}{18})^5} = \left( \dfrac{\frac{7}{-5}}{-\frac{14}{18}} \right)^5 = \left( \frac{7}{-5} \cdot \frac{18}{-14} \right)^5 = \left( \frac{7 \cdot 18}{-5 \cdot -14} \right)^5 = \left( \frac{126}{70} \right)^5 = \left(\frac{9}{5}\right)^5$. c) $\frac{(\frac{1}{7})^{2018}}{(\frac{1}{7})^{2018}} = 1$. d) $\frac{6^6 + 6^3 3^3 + 3^6}{-73}$. Note $6^3 = 216$, $3^3 = 27$, $6^6 = (6^3)^2 = 216^2 = 46656$, $3^6 = (3^3)^2 = 27^2 = 729$. So numerator $= 46656 + 216 \cdot 27 + 729 = 46656 + 5832 + 729 = 53217$. Then $\frac{53217}{-73} = -729$. 4. **Solve Bài 20:** a) $\frac{(-\frac{5}{4})^2}{(\frac{35}{24})^2} = \left(\frac{-\frac{5}{4}}{\frac{35}{24}}\right)^2 = \left(-\frac{5}{4} \cdot \frac{24}{35}\right)^2 = \left(-\frac{120}{140}\right)^2 = \left(-\frac{6}{7}\right)^2 = \frac{36}{49}$. b) $(-\frac{1}{2})^2 \cdot (\frac{2}{5})^2 = \frac{1}{4} \cdot \frac{4}{25} = \frac{1}{25}$. c) $(\frac{1}{9})^2 \cdot (\frac{1}{3})^3 = \frac{1}{81} \cdot \frac{1}{27} = \frac{1}{2187}$. d) $(-\frac{1}{2})^3 \cdot (\frac{3}{2})^3 = \left(-\frac{1}{2} \cdot \frac{3}{2}\right)^3 = \left(-\frac{3}{4}\right)^3 = -\frac{27}{64}$. 5. **Solve Bài 21:** a) $(\frac{2}{5} + \frac{1}{3})^2 = (\frac{6}{15} + \frac{5}{15})^2 = (\frac{11}{15})^2 = \frac{121}{225}$. b) $(-\frac{20}{3})^3 \cdot (-\frac{18}{5})^2 = (-1)^3 (\frac{20}{3})^3 \cdot (\frac{18}{5})^2 = - \frac{8000}{27} \cdot \frac{324}{25} = - \frac{8000 \cdot 324}{27 \cdot 25} = - \frac{2,592,000}{675} = -3840$. c) $A = (3^2)^2 - (-2^3)^2 - (-5^2)^2 = 3^{4} - (-8)^2 - (-25)^2 = 81 - 64 - 625 = -608$. d) $B = 2^3 + 3 \cdot (\frac{1}{2})^0 \cdot (\frac{1}{2})^2 \cdot 4 + [(-2)^2 : \frac{1}{2}] : 8 = 8 + 3 \cdot 1 \cdot \frac{1}{4} \cdot 4 + [4 : \frac{1}{2}] : 8 = 8 + 3 + 8 : 8 = 8 + 3 +1 = 12$. e) $A=3^2 \cdot \frac{1}{243} \cdot 81^2 \cdot \frac{1}{3^3} = 9 \cdot 3^{-5} \cdot (3^4)^2 \cdot 3^{-3} = 9 \cdot 3^{-5} \cdot 3^{8} \cdot 3^{-3} = 3^{2 - 5 + 8 -3} = 3^{2} = 9$. 6. **Solve Bài 22:** a) $A = (-\frac{1}{3})^3 \cdot (-\frac{1}{3})^2 \cdot (-\frac{1}{3}) = (-\frac{1}{3})^{3+2+1} = (-\frac{1}{3})^{6} = \frac{1}{3^6} = \frac{1}{729}$. b) $B = (-\frac{1}{3})^{-1} - (-\frac{6}{7})^0 + \frac{(1/2)^2}{2} = -3 - 1 + \frac{1/4}{2} = -3 -1 + \frac{1}{8} = -4 + \frac{1}{8} = -\frac{31}{8}$. c) $C = (0.1^2)^0 + [(\frac{1}{7})^2 \cdot \frac{1}{49} \cdot ((2^3)^2 : 2^{5})] = 1 + [\frac{1}{49} \cdot \frac{1}{49} \cdot (8^2 : 32)] = 1 + [\frac{1}{2401} \cdot (64 : 32)] = 1 + [\frac{1}{2401} \cdot 2] = 1 + \frac{2}{2401} \approx 1.00083$. d) $B = \frac{(-0.5)^5}{(-0.5)^3} - \frac{(17/2)^7}{(17/2)^6} = (-0.5)^{5-3} - (17/2)^{7-6} = (-0.5)^2 - (17/2)^1 = 0.25 - 8.5 = -8.25$. e) $A = (1 \frac{3}{4})^3 - (1 \frac{3}{4})^2 + (-1.031)^0 = (\frac{7}{4})^3 - (\frac{7}{4})^2 + 1 = \frac{343}{64} - \frac{49}{16} + 1 = \frac{343}{64} - \frac{196}{64} + \frac{64}{64} = \frac{211}{64}$. f) $B = (\frac{2}{3})^3 - 4 \cdot (-\frac{7}{4})^2 + (\frac{2}{3})^3 = 2 \cdot (\frac{8}{27}) - 4 \cdot \frac{49}{16} = \frac{16}{27} - \frac{196}{16} = \frac{16}{27} - 12.25 = -11.63$ approx. 7. **Solve Bài 23:** a) $\frac{2^{15} \cdot 9^{4}}{6^{6} \cdot 8^{3}} = \frac{2^{15} \cdot (3^{2})^{4}}{(2 \cdot 3)^{6} \cdot (2^{3})^{3}} = \frac{2^{15} \cdot 3^{8}}{2^{6} \cdot 3^{6} \cdot 2^{9}} = \frac{2^{15} \cdot 3^{8}}{2^{15} \cdot 3^{6}} = 3^{2} = 9$. b) $\frac{(-0.3)^2 \cdot 8}{(0.6)^7} = \frac{0.09 \cdot 8}{0.6^7} = \frac{0.72}{0.02799} \approx 25.72$. c) $\frac{45^{10} \cdot 5^{20}}{75^{15}} = \frac{(9\cdot5)^{10} \cdot 5^{20}}{(3 \cdot 5)^{15}} = \frac{9^{10} \cdot 5^{10} \cdot 5^{20}}{3^{15} \cdot 5^{15}} = \frac{9^{10} \cdot 5^{30}}{3^{15} \cdot 5^{15}} = 9^{10} \cdot \frac{5^{15}}{3^{15}} = (3^2)^{10} \cdot \left(\frac{5}{3}\right)^{15} = 3^{20} \cdot \left(\frac{5}{3}\right)^{15}$. d) $\frac{3^7 \cdot 16^3}{12^5 \cdot 27^2} = \frac{3^7 \cdot (2^4)^3}{(2^{2} \cdot 3)^5 \cdot (3^{3})^{2}} = \frac{3^7 \cdot 2^{12}}{2^{10} \cdot 3^{5} \cdot 3^{6}} = \frac{3^{7} \cdot 2^{12}}{2^{10} \cdot 3^{11}} = 2^{2} \cdot 3^{-4} = \frac{4}{81}$. e) $\frac{2^3 \cdot (0.5)^3 \cdot 3^7}{2 \cdot (0.5)^4 \cdot 3^8} = \frac{2^{3} \cdot 2^{-3} \cdot 3^{7}}{2^{1} \cdot 2^{-4} \cdot 3^{8}} = \frac{2^{0} \cdot 3^{7}}{2^{-3} \cdot 3^{8}} = 2^{3} \cdot 3^{-1} = \frac{8}{3}$. f) $(0.8)^5 = \text{approximately } 0.32768$. g) $\frac{3^7 \cdot 81}{27^{0} \cdot 9^5} = \frac{3^7 \cdot 3^4}{1 \cdot 3^{10}} = 3^{7+4-10} = 3^{1} = 3$. h) $\frac{9^{2} \cdot 2^{11}}{16^{2} \cdot 6^{3}} = \frac{3^{4} \cdot 2^{11}}{2^{8} \cdot (2 \cdot 3)^{3}} = \frac{3^{4} \cdot 2^{11}}{2^{8} \cdot 2^{3} \cdot 3^{3}} = \frac{3^{4} \cdot 2^{11}}{2^{11} \cdot 3^{3}} = 3^{4-3} = 3$. i) $A = \frac{(-3)^{10} \cdot 15^{5}}{9^{4}} = \frac{3^{10} \cdot (3 \cdot 5)^5}{3^{8}} = \frac{3^{10} \cdot 3^{5} \cdot 5^{5}}{3^{8}} = 3^{10+5-8} \cdot 5^{5} = 3^{7} \cdot 5^{5}$. k) $B = \frac{4^{30} \cdot 3^{43}}{2^{57} \cdot 27^{15}} = \frac{(2^2)^{30} \cdot 3^{43}}{2^{57} \cdot (3^3)^{15}} = \frac{2^{60} \cdot 3^{43}}{2^{57} \cdot 3^{45}} = 2^{60 - 57} \cdot 3^{43 - 45} = 2^3 \cdot 3^{-2} = \frac{8}{9}$. **Final answers:** a) $5^{22}$ b) $2^{17}$ c) $5^7$ d) $3^5$ e) $5^8$ f) $2^9$ Bài 19: a) $\frac{16^3}{81^3}$, b) $\left(\frac{9}{5}\right)^5$, c) 1, d) -729 Bài 20: a) $\frac{36}{49}$, b) $\frac{1}{25}$, c) $\frac{1}{2187}$, d) $-\frac{27}{64}$ Bài 21: a) $\frac{121}{225}$, b) $-3840$, c) $-608$, d) 12, e) 9 Bài 22: a) $\frac{1}{729}$, b) $-\frac{31}{8}$, c) $\approx 1.00083$, d) $-8.25$, e) $\frac{211}{64}$, f) $\approx -11.63$ Bài 23: a) 9, b) $\approx 25.72$, c) $3^{20} \cdot \left(\frac{5}{3}\right)^{15}$, d) $\frac{4}{81}$, e) $\frac{8}{3}$, f) $\approx 0.32768$, g) 3, h) 3, i) $3^{7} \cdot 5^{5}$, k) $\frac{8}{9}$