a) [tex]([/tex][tex]\frac{1}{3}[/tex][tex])^{5}[/tex] × [tex]6^{5}[/tex] = [tex](\frac{1}{3}[/tex] × [tex]6)^{5}[/tex] = [tex]2^{5}[/tex] = 32
b) [tex]21^{3}[/tex] ÷ [tex]7^{3}[/tex] = [tex](21 : 7)^{3}[/tex]= [tex]3^{3}[/tex] = 27
c) [tex]((\frac{1}{2}) ^{4}) ^{2}[/tex] × [tex]2^{8}[/tex] = [tex](\frac{1}{2}) ^{8}[/tex] × [tex]2^{8}[/tex] = [tex](\frac{1}{2}[/tex] × [tex]2)^{8}[/tex] = [tex]1^{8}[/tex] = 1
d) ( [tex]2^{11}[/tex] ÷ [tex]2^{3}[/tex]) ÷ [tex](2^{4}) ^{2}[/tex] = [tex]2^{8}[/tex] ÷ [tex]2^{8}[/tex] = [tex]2^{0}[/tex] = 1
e) [tex]\frac{(7^{6} * 7^{4}) ^{3} }{(7^{12} : 7^{8} ) ^{7} }[/tex] = [tex]\frac{(7^{10}) ^{3} }{(7^{4}) ^{7} }[/tex] = [tex]\frac{7^{30} }{7^{28} }[/tex] = [tex]7^{30}[/tex] : [tex]7^{28}[/tex] = [tex]7^{2}[/tex] = 49
f) [tex](0,5) ^{6}[/tex] × [tex]4^{3}[/tex] = [tex](\frac{1}{2} )^{6}[/tex] × [tex](2^{2} )^{3}[/tex] = [tex](\frac{1}{2} )^{6}[/tex] × [tex]2^{6}[/tex] = [tex](\frac{1}{2} * 2 )^{6}[/tex] = [tex]1^{6}[/tex] = 1
g) [tex]8^{5}[/tex] : [tex]4^{7}[/tex] = [tex](2^{3} )^{5}[/tex] : [tex](2^{2} )^{7}[/tex] = [tex]2^{15}[/tex] : [tex]2^{14}[/tex] = [tex]2^{1}[/tex] = 2
h) [tex]\frac{16^{3} }{8^{4} }[/tex] = [tex]\frac{(2^{4} )^{3} }{(2^{3} )^{4} }[/tex] = [tex]\frac{2^{12} }{2^{12} }[/tex] = [tex]2^{0}[/tex] = 1
i) [tex]\frac{125^{7} }{25^{10} }[/tex] = [tex]\frac{(5^{3} )^{7} }{(5^{2} )^{10} }[/tex] = [tex]\frac{5^{21} }{5^{20} }[/tex] = [tex]5^{21}[/tex] ÷ [tex]5^{20}[/tex] = [tex]5^{1}[/tex] = 5
j) [tex]\frac{9^{7} * 27^{5} }{81^{7} }[/tex] = [tex]\frac{(3^{2} )^{7} * (3^{3} )^{5} }{(3^{4} )^{7} }[/tex] = [tex]\frac{3^{14} * 3^{15} }{3^{28} }[/tex] = [tex]\frac{3^{29} }{3^{28}}[/tex] = [tex]3^{29}[/tex] : [tex]3^{28}[/tex] = [tex]3^{1}[/tex] = 3
k) [tex]125^{5}[/tex] : [tex](25^{4} * 5^{6})[/tex] = [tex](5^{3} )^{5}[/tex] : [tex]((5^{2} )^{4} * 5^{6})[/tex] = [tex]5^{15}[/tex] : [tex](5^{8}[/tex] × [tex]5^{6})[/tex] = [tex]5^{15}[/tex] : [tex]5^{14}[/tex] = [tex]5^{1}[/tex] = 5
