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\(\frac{2^7.9^3}{6^5.8^2}=\frac{2^7.\left(3^2\right)^3}{\left(2.3\right)^5.\left(2^3\right)^2}=\frac{2^7..3^6}{2^5.3^5.2^6}=\frac{2^7.3^6}{2^{11}.3^5}=\frac{3}{2^4}=\frac{3}{16}\)
\(=\dfrac{1}{2}\cdot\dfrac{6}{5}\cdot\dfrac{8}{7}+\dfrac{3}{7}\cdot\dfrac{9}{5}=\dfrac{24}{35}+\dfrac{27}{35}=\dfrac{51}{35}\)
a/\(\frac{4^5.9^4-2.6^9}{2^{10}.3^8+6^8.20}=\frac{\left(2^2\right)^5.\left(3^2\right)^4-2.\left(2.3\right)^9}{2^{10}.3^8+\left(2.3\right)^8.2^2.5}\)
= \(\frac{2^{10}.3^8-2.2^9.3^9}{2^{10}.3^8+2^8.3^8.2^2.5}=\frac{2^{10}.3^8-2^{10}.3^9}{2^{10}.3^8+2^{10}.3^8.5}=\frac{2^{10}.3^8.\left(1-3\right)}{2^{10}.3^8\left(1+5\right)}=\frac{1-3}{1+5}=\frac{-2}{6}=-3\)
\(a\)) \(\left(4^2.4^3\right):2^{10}\)
\(=4^5:2^{10}\)
\(=\left(2^2\right)^5:2^{10}\)
\(=2^{10}:2^{10}\)
\(=1\)
\(b\)) \(\left(0,6\right)^5:\left(0,2\right)^6\)
\(=\left(\frac{3}{5}\right)^5:\left(\frac{1}{5}\right)^6\)
\(=\frac{3^5}{5^5}:\frac{1^6}{5^6}\)
\(=\frac{3^5}{5^5}.5^6\)
\(=\frac{3^5.5^6}{5^5}\)
\(=3^5.5\)
\(=243.5\)
\(=1215\)
\(c\)) \(\left(2^7.9^3\right):\left(6^5.8^2\right)\)
\(=\left[2^7.\left(3^2\right)^3\right]:\left[\left(2.3\right)^5.\left(2^4\right)^2\right]\)
\(=\left(2^7.3^6\right):\left[2^5.3^5.2^8\right]\)
\(=\left(2^7.3^6\right).\left(\frac{1}{2^5.3^5.2^8}\right)\)
\(=\frac{2^7.3^6.1}{\left(2^5.2^8\right).3^5}\)
\(=\frac{2^7.3^6}{2^{13}.3^5}\)
\(=\frac{3}{2^6}\)
\(=\frac{3}{64}\)