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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) \(\frac{34^5.2^5}{2^{14}.17^5}=\frac{2^5.17^5.2^5}{2^{14}.17^5}=\frac{2^{10}}{2^{14}}=\frac{1}{2^4}=\frac{1}{16}\)
=\(\frac{5^4.20^5}{25^4.4^3}\)
=\(\frac{5^4.\left(2^2.5\right)^5}{\left(5^2\right)^4.\left(2^2\right)^3}\)
=\(\frac{5^4.2^{10}.5^5}{5^8.2^6}\)
=\(\frac{5^9.2^{10}}{5^8.2^6}\)
=\(\frac{5.2^4}{1.1}\)
=\(\frac{80}{1}\)
=80
\(\dfrac{4^3\cdot4^4}{2^{13}}=\dfrac{2^6\cdot2^8}{2^{13}}=2\)