a. \(\dfrac{5^3.90.4^3}{25^2.3^2.2^{13}}=\dfrac{5^3.5.3^2.2.\left(2^2\right)^3}{\left(5^2\right)^2.3^2.2^{13}}=\dfrac{5^4.3^2.2^7}{5^4.3^2.2^{13}}=\dfrac{1}{2^6}=\dfrac{1}{64}\)
b. \(\dfrac{15^2.16^4-15^3.16^3}{12^2.20^3-20^2.12^3}=\dfrac{15^2.16^3\left(16-15\right)}{12^2.20^2\left(20-12\right)}=\dfrac{15^2.16^3}{12^2.20^2.8}=\dfrac{\left(3.5\right)^2.\left(2^4\right)^3}{\left(3.2^2\right)^2.\left(2^2.5\right)^2.2^3}=\dfrac{3^2.5^2.2^{12}}{3^2.2^4.2^4.5^2.2^3}=\dfrac{3^2.5^2.2^{12}}{3^2.5^2.2^{11}}=\dfrac{1}{2}\)
\(a\))\(\dfrac{5^3.90.4^3}{25^2.3^2.2^{13}}\)
\(=\dfrac{5^3.3^2.10.\left(2^2\right)^3}{\left(5^2\right)^2.3^2.2^{13}}\Leftrightarrow\dfrac{1.1.10.1}{5.1.2^7}=\dfrac{1}{64}\)
a)\(\dfrac{5^{3^{ }}.90.4^3}{25^{2^{ }}3^2.2^{13}}=\dfrac{5^3.2.5.3^{2^{ }}.2^6}{5^4.3^2.2^{10}}=\dfrac{5^4.2^7.3^2}{5^4.3^2.2^{10}}=\dfrac{2^7}{2^{10}}=\dfrac{1}{8}\)
b)\(\dfrac{15^2.16^4-15^3.16^3}{12^2.20^3-20^2.12^3}=\dfrac{15^2.16^3.16-15^2.15.16^3}{12^2.20^2.20-20^212^2.12}=\dfrac{15^2.16^3.\left(16-15\right)}{12^2.20^2.\left(20-12\right)}=\dfrac{15^2.16^3}{12^2.20^2.8}=\dfrac{\left(3.5\right)^2.\left(4.4\right)^3}{\left(3.4\right)^2.\left(4.5\right)^2.2^3}=\dfrac{3^2.5^2.2^{12}}{3^2.4^2.4^2.5^2.2^3}=\dfrac{2^{12}}{4^{4^{ }}.2^3}=\dfrac{2^{12}}{2^8.2^3}=\dfrac{2^{12}}{2^{11}}=2\)
