Verified answer

1)

[tex](a^{4} )^{2} : (a^{2} )^{3}=a^{8}:a^{6} =a^{2} \\a^{4} a^{6} :a^{9} =a^{4+6-9} =a\\(a^{4} )^{2} :(a^{2} )^{3} =a^{8} :a^{6} =a^{2}[/tex]
2)
[tex]\frac{3^{10} *(3^{5} )^{3} }{(3^{5} )^{4}*3 } =\frac{3^{10} *3^{15} }{3^{20}*3 }=\frac{3^{25} }{3^{21} }=3^{4}[/tex]
[tex]\frac{4^{3}*16^{2} }{2^{12} } =\frac{2^{6}*2^{8} }{2^{12} }=\frac{2^{8} }{2^{6} } =2^{2}[/tex]

[tex]\frac{5^{2}*625 }{125^{2} } =\frac{5^{2}*5^{4} }{5^{6}}= \frac{5^{4} }{5^{4}}=1[/tex]
3)
[tex]-14b^{2} c^{8} d^{9} *1\frac{2}{7} b^{6} d^{3} c=-14b^{2} c^{8} d^{9} *\frac{9}{7} b^{6} d^{3} c=-2b^{2} c^{8} d^{9} *9 b^{6} d^{3} c=-18b^{8} c^{9} d^{12}[/tex]

[tex]\frac{4}{9}a^{4}c*(-12a^{2}c^{3} ) * 1,8a^{4} b^{5} =\frac{4}{9}a^{4}c*(-12a^{2}c^{3} ) * \frac{9}{5} a^{4} b^{5} =-4a^{4}c*12a^{2}c^{3} * \frac{1}{5} a^{4} b^{5}= -\frac{48}{5} a^{10} b^{5} c^{4}[/tex]
[tex]-6\frac{1}{2}mn^{8} p^{11} *3\frac{5}{13} m^{5} n^{5} =-\frac{13}{2}mn^{8} p^{11} *\frac{44}{13} m^{5} n^{5}=-mn^{8} p^{11} *22m^{5} n^{5}=-22 m^{6} n^{13} p^{11}[/tex]