[tex]b) \sqrt[4]{625 \times 81} = \sqrt[4]{625} \times \sqrt[4]{81} [/tex]
[tex] = \sqrt[4]{ {25}^{2} } \times \sqrt[4]{ {9}^{2} } [/tex]
[tex] = \sqrt[4]{ {( {5}^{2}) }^{2} } \times \sqrt[4]{ { ({3}^{2}) }^{2} } [/tex]
[tex] = \sqrt[4]{ {5}^{2 \times 2} } \times \sqrt[4]{ {3}^{2 \times 2} } [/tex]
[tex] = \sqrt[4]{ {5}^{4} } \times \sqrt[4]{ {3}^{4} } [/tex]
[tex] = {5}^{ \frac{4}{4} } \times {3}^{ \frac{4}{4} } [/tex]
[tex] = {5}^{1} \times {3}^{1} [/tex]
[tex] = 5 \times 3[/tex]
[tex] = 15[/tex]
[tex]c) \sqrt[5]{ {2}^{10} \times {3}^{ - 5} } = \sqrt[5]{ {2}^{10} } \times \sqrt[5]{ {3}^{ - 5} } [/tex]
[tex] = {2}^{ \frac{10}{5} } \times {3}^{ - \frac{5}{5} } [/tex]
[tex] = {2}^{2} \times {3}^{ - 1} [/tex]
[tex] = 4 \times \frac{1}{3} [/tex]
[tex] = \frac{4 \times 1}{3} [/tex]
[tex] = \frac{4}{3} [/tex]
[tex]d) \sqrt[4]{256 \times 4096} = \sqrt[4]{256} \times \sqrt[4]{4096} [/tex]
[tex] = \sqrt[4]{ {16}^{2} } \times \sqrt[4]{ {64}^{2} } [/tex]
[tex] = \sqrt[4]{ { ({4}^{2}) }^{2} } \times \sqrt[4]{ { ({8}^{2} )}^{2} } [/tex]
[tex] = \sqrt[4]{ {4}^{2 \times 2} } \times \sqrt[4]{ {8}^{2 \times 2} } [/tex]
[tex] = \sqrt[4]{ {4}^{4} } \times \sqrt[4]{ {8}^{4} } [/tex]
[tex] = {4}^{ \frac{4}{4} } \times {8}^{ \frac{4}{4} } [/tex]
[tex] = {4}^{1} \times {8}^{1} [/tex]
[tex] = 4 \times 8[/tex]
[tex] = 32[/tex]
[tex]e) \sqrt[21]{128 \times {3}^{14} } = \sqrt[21]{128} \times \sqrt[21]{ {3}^{14} } [/tex]
[tex] = \sqrt[21]{ {2}^{7} } \times {3}^{ \frac{14}{21} } [/tex]
[tex] = \sqrt[21]{{2}^{7} } \times {3}^{ \frac{2}{3} } [/tex]
[tex] = {2}^{ \frac{7}{21} } \times {3}^{ \frac{2}{3} } [/tex]
[tex]={2}^{\frac{1}{3}}\times{3}^{\frac{2}{3}}[/tex]
[tex]f) \sqrt[12]{ {343}^{3} \times {49}^{5} \times {7}^{5} } = \sqrt[12]{ {343}^{3} } \times \sqrt[12]{ {49}^{5} } \times \sqrt[12]{ {7}^{5} } [/tex]
[tex] = \sqrt[12]{ {( {7}^{3}) }^{3} } \times \sqrt[12]{ { ({7}^{2} )}^{5} } \times \sqrt[12]{ {7}^{5} } [/tex]
[tex] = \sqrt[12]{ {7}^{3 \times 3} } \times \sqrt[12]{ {7}^{2 \times 5} } \times \sqrt[12]{ {7}^{5} } [/tex]
[tex] = \sqrt[12]{ {7}^{9} } \times \sqrt[12]{ {7}^{10} } \times \sqrt[12]{ {7}^{5} } [/tex]
[tex] = {7}^{ \frac{9}{12} } \times {7}^{ \frac{10}{12} } \times {7}^{ \frac{5}{12} } [/tex]
[tex] = {7}^{ \frac{9}{12} + \frac{10}{12} + \frac{5}{12} } [/tex]
[tex] = {7}^{ \frac{9 + 10 + 5}{12} } [/tex]
[tex] = {7}^{ \frac{24}{12} } [/tex]
[tex] = {7}^{2} [/tex]
[tex] = 49[/tex]
