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a) \(\frac{75^3.3^7}{81^4.5^6}=\frac{5^3.3^3.5^3.3^7}{\left(3^4\right)^4.5^6}=\frac{5^6.3^3.3^7}{3^{16}.5^6}=\frac{3^{10}}{3^{16}}=\frac{1}{3^6}=\frac{1}{729}\)
b) \(\frac{6^6.4^2}{3^{12}.2^8}=\frac{2^6.3^6.\left(2^2\right)^2}{3^{12}.2^8}=\frac{2^6.3^6.2^4}{3^{12}.2^8}=\frac{2^{10}.3^6}{3^{12}.2^8}=\frac{2^2.1}{3^6}=\frac{4}{729}\)
c) \(\frac{34^5.2^5}{2^{14}.17^5}=\frac{2^5.17^5.2^5}{2^{14}.17^5}=\frac{2^{10}}{2^{14}}=\frac{1}{2^4}=\frac{1}{16}\)
\(\dfrac{27^4\cdot\left(-8\right)^2}{9^5\cdot6^3}=\dfrac{\left(3^3\right)^4\cdot\left(2^3\right)^2}{\left(3^2\right)^5\cdot\left(2\cdot3\right)^3}=\dfrac{3^{12}\cdot2^6}{3^{10}\cdot2^3\cdot3^3}=\dfrac{2^3}{3}=\dfrac{8}{3}\)
\(\dfrac{24^4.\left(-8\right)^2}{9^5.6^3}\)
=\(\dfrac{\left(4.6\right)^4.\left(2^3\right)^2}{\left(3^2\right)^5.6^3}\)
=\(\dfrac{4^4.6^4.2^6}{3^{10}.6^3}\)
=\(\dfrac{2^{14}.6}{3^{10}}\)
\(\dfrac{4^{^5}.9^{^4}-2.6^{^9}}{2^{^{10}}.3^{^8}+6^{^8}.20}\)=\(\dfrac{2^{^{10}}.3^{^8}-2.2^{^9}.3^{^9}}{2^{^{10}}.3^{^8}+2^{^8}.3^{^8}.2^{^2}.5}\)=\(\dfrac{2^{^{10}}.3^{^8}-2^{^{10}}.3^{^9}}{2^{^{10}}.3^{^8}+2^{^{10}}.3^{^8}.5}\)=\(\dfrac{2^{^{10}}.3^{^8}\left(1-3\right)}{2^{^{10}}.3^{^8}\left(1+5\right)}\)=\(\dfrac{1-3}{1+5}\)=\(\dfrac{1}{3}\)
45.94−2.69210.38+68.2045.94−2.69210.38+68.20=210.38−2.29.39210.38+28.38.22.5210.38−2.29.39210.38+28.38.22.5=210.38−210.39210.38+210.38.5210.38−210.39210.38+210.38.5=210.38(1−3)210.38(1+5)210.38(1−3)210.38(1+5)=1−31+51−31+5=13