\[\boxed{\text{542\ (542).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\textbf{а)}\ 4^{5} \cdot {2,5}^{5} = (4 \cdot 2,5)^{5} = 10^{5} =\]
\[= 100\ 000\]
\[\textbf{б)}\ \left( \frac{1}{3} \right)^{13} \cdot 3^{13} = \left( \frac{1}{3} \cdot 3 \right)^{13} =\]
\[= 1^{13} = 1\]
\[\textbf{в)}\ {0,2}^{9} \cdot 5^{7} = {0,2}^{2} \cdot (0,2 \cdot 5)^{7} =\]
\[= {0,2}^{2} \cdot 1^{7} = 0,04\]
\[\textbf{г)}\ {0,4}^{10} \cdot {2,5}^{12} =\]
\[= (0,4 \cdot 2,5)^{10} \cdot {2,5}^{2} =\]
\[= 1^{10} \cdot 6,25 = 6,25\]
\[\textbf{д)}\ {0,2}^{6} \cdot 25^{3} = {0,2}^{6} \cdot \left( 5^{2} \right)^{3} =\]
\[= (0,2 \cdot 5)^{6} = 1^{6} = 1\]
\[\textbf{е)}\ \left( \frac{1}{9} \right)^{6} \cdot 81^{4} = \left( \frac{1}{9} \right)^{6} \cdot 9^{8} =\]
\[= 9^{2} \cdot \left( \frac{1}{9} \cdot 9 \right)^{6} = 9^{2} \cdot 1^{6} = 81\ \]