[tex]36^4 : 12^3 =\dfrac{36^4}{12^3}=\dfrac{36\cdot36^3}{12^3}=36\cdot\dfrac{36^3}{12^3}=36\cdot\left(\dfrac{36}{12}\right)^3=36\cdot3^3=36\cdot27=972[/tex]
[tex]50^5 : 10^4 =\dfrac{50^5}{10^4}=\dfrac{50\cdot50^4}{10^4}=50\cdot\left(\dfrac{50}{10}\right)^4=50\cdot5^4=50\cdot625=31250[/tex]
[tex]100^2\cdot\left(\frac12\right)^4 =(10^2)^2\cdot\left(\frac12\right)^4 =1\big0^4\cdot\left(\frac12\right)^4 =\left(10\cdot\frac12\right)^4=5^4=625[/tex]
[tex]\dfrac{81^{2}\cdot16^{4} }{(3\cdot2)^{6} }=\dfrac{(3^4)^{2}\cdot(2^4)^{4} }{3^6\cdot2^{6} }=\dfrac{3^8\cdot2^{16} }{3^6\cdot2^{6} }=3^2\cdot2^{10}=9\cdot1024=9216[/tex]
[tex]\dfrac{-32^{4}\cdot(-4)^{0}\cdot5^{2}}{(2^{4})^{5}\cdot125}= \dfrac{-(2^5)^{4}\cdot1\cdot5^{2}}{2^{20}\cdot5^3}= -\dfrac{2^{20}\cdot5^{2}}{2^{20}\cdot5^3}=-\dfrac15[/tex]
[tex]\dfrac{9^{2}\cdot(-18)^{0}\cdot27^{4}}{81\cdot(-9)^{4}}= \dfrac{9^{2}\cdot1\cdot27^{4}}{81\cdot9^{4}}= \dfrac{(3^2)^{2}\cdot(3^3)^{4}}{3^4\cdot(3^2)^{4}}= \dfrac{3^4\cdot3^{12}}{3^4\cdot3^8}=3^4=81[/tex]
