Rút gọn phân số sau thành phân số tối giản

a) \(\dfrac{5^3.90.4^3}{25^2.3^2.2^{13}}\) = \(\dfrac{5^3.3^2.2.5.2^6}{5^4.3^2.2^{13}}\) = \(\dfrac{5^4.3^2.2^7}{5^4.3^2.2^{13}}\) = \(\dfrac{2^7}{2^{13}}\) = \(\dfrac{1}{2^6}\) = \(\dfrac{1}{64}\)

b) \(\dfrac{18.27+18.\left(-23\right)}{34.4-4.52}\) = \(\dfrac{18\left(27-23\right)}{4\left(34-52\right)}\) = \(\dfrac{18.4}{4.\left(-18\right)}\) = \(\dfrac{18}{-18}\) = -1

c) \(\dfrac{15^2.16^4-15^3.16^3}{12^2.20^3-20^2.12^3}\) = \(\dfrac{15^2.16^3.16-15^2.15.16^3}{12^2.20^2.20-20^2.12^2.12}\) = \(\dfrac{15^2.16^3.\left(16-15\right)}{12^2.20^2.\left(20-12\right)}\)= \(\dfrac{15^2.16^3}{12^2.20^2.8}\) = \(\dfrac{\left(3.5\right)^2.\left(2^4\right)^3}{\left(3.4\right)^2.\left(4.5\right)^2.2^3}\) = \(\dfrac{3^2.5^2.2^{12}}{3^2.4^2.4^2.5^2.2^3}\) = \(\dfrac{2^{12}}{4^4.2^3}\) = \(\dfrac{2^{12}}{2^8.2^3}\) = \(\dfrac{2^{12}}{2^{11}}\) = 2

d) \(\dfrac{2.3+4.6+14.21}{3.5+6.10+21.35}\) = \(\dfrac{2.3+2^2.2.3+2.7.3.7}{3.5+3.2.5.2+3.7.5.7}\) = \(\dfrac{2.3+2^3.3+2.3}{3.5+3.5.2^2+3.5}\) = \(\dfrac{2.3 \left(1+2^2+1\right)}{3.5\left(1+2^2+1\right)}\) = \(\dfrac{2.3}{3.5}\) = \(\dfrac{2}{5}\)