a)
[tex] {5}^{3} \times 5 \div {5}^{4} = {5}^{4} \div {5}^{4} = {5}^{0} = 1[/tex]
b)
[tex] \frac{ {5}^{13} \times {5}^{12} }{ {5}^{6} \times 5} = \frac{ {5}^{25} }{ {5}^{7} } = {5}^{18} [/tex]
c)
[tex] {13}^{4} \times {15}^{4} = ({13 \times 15})^{4} = {195}^{4} [/tex]
d)
[tex] {3}^{8} \times {4}^{4} = {3}^{4} \times {3}^{4} \times {4}^{4} = ( {3 \times 3 \times 4)}^{4} = {36}^{4} [/tex]
e)
[tex] {81}^{12} \times {5}^{4} = ( {9}^{2} )^{12} \times {5}^{4} = ( {3}^{2} )^{2} )^{12} \times {5}^{4} = {3}^{48} \times {5}^{4} = {3}^{24} \times {3}^{24} \times {5}^{4} = {3}^{12} \times {3}^{12} \times {3}^{12} \times {3}^{12} \times {5}^{4} = {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {3}^{4} \times {5}^{4} = (3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5)^{4} = (9 \times 9 \times 9 \times 9 \times 9 \times 9 \times 5)^{4} = (81 \times 81 \times 81 \times 5)^{4} = {2657205}^{4} [/tex]
f)
[tex] {625}^{4} \times {5}^{4} = ( {5}^{3} )^{4} \times {5}^{4} = {5}^{12} \times {5}^{4} = {5}^{16} [/tex]
