a)
[tex]{2}^{2} \times {3}^{4} = {2}^{2} \times {( {3}^{2} )}^{2} = {2}^{2} \times {9}^{2} = {(2 \times 9)}^{2}[/tex]
[tex] = {18}^{2} = 324[/tex]
[tex] {2}^{2} \times {3}^{4} \times {5}^{6} = {2}^{2} \times {( {3}^{2} )}^{2} \times {( {5}^{3} )}^{2} [/tex]
[tex] = {2}^{2} \times {9}^{2} \times {125}^{2} = {(2 \times 9 \times 125)}^{2} [/tex]
[tex] = {(18 \times 125)}^{2} = {(2250)}^{2} [/tex]
[tex] {2}^{100} : {2}^{84} = {2}^{100 - 84} = {2}^{16} = {( {2}^{8} )}^{2} = {256}^{2} [/tex]
[tex] {2}^{6} \times {5}^{2} = {( {2}^{3} )}^{2} \times {5}^{2} = {8}^{2} \times {5}^{2} = {(8 \times 5)}^{2}[/tex]
[tex] = {40}^{2} = 1600[/tex]
b)
[tex] \sqrt{ {14}^{2} } = 14[/tex]
[tex] \sqrt{ {7}^{4} } = \sqrt{ {( {7}^{2} )}^{2} } = {7}^{2} = 49[/tex]
[tex] \sqrt{ {a}^{2} } = a[/tex]
c)
[tex] \sqrt{{2}^{2} \times {3}^{4}} = \sqrt{ {18}^{2} } = 18[/tex]
[tex] \sqrt{{2}^{2} \times {3}^{4} \times {5}^{6}} = \sqrt{ {2250}^{2} } = 2250[/tex]
[tex] \sqrt{{2}^{100} : {2}^{84}} = \sqrt{ {256}^{2} } = 256[/tex]
[tex] \sqrt{{2}^{6} \times {5}^{2}} = \sqrt{ {40}^{2} } = 40[/tex]
