[tex]1) \sqrt{16 \times {3}^{4} } = \sqrt{16} \sqrt{ {3}^{4} } = \sqrt{ {4}^{2} } \times {3}^{2} = 4 \times {3}^{2} = 4 \times 9 = 36 [/tex]
[tex]2) \sqrt[12]{ {8}^{2} \times {5}^{6} } = \sqrt[12]{64 \times {5}^{6} } = \sqrt[12]{ {10}^{6} } = \sqrt{ |10| } = \sqrt{10} [/tex]
[tex]3) \sqrt[5]{ {3}^{10} \times {2}^{ - 5} } = \sqrt[5]{ {3}^{10} } \sqrt[5]{ {2}^{ - 5} } = {3}^{2} \times {2}^{ - 1} = 9 \times \frac{1}{2} = \frac{9}{1} \times \frac{1}{2} = \frac{9 \times 1}{1 \times 2} = \frac{9}{2} [/tex]
[tex]4) \sqrt[4]{81 \times \frac{1}{256} } = \sqrt[4]{81} \sqrt[4]{ \frac{1}{256} } = \sqrt[4]{ {3}^{4} } \times \frac{ \sqrt[4]{1} }{ \sqrt[4]{256} } = 3 \times \frac{1}{ \sqrt[4]{ {4}^{4} } } = 3 \times \frac{1}{4} = \frac{3}{1} \times \frac{1}{4} = \frac{3 \times 1}{1 \times 4} = \frac{3}{4} [/tex]
[tex]5) \sqrt[3]{3 \times \sqrt{81} } = \sqrt[3]{3 \times 9} = \sqrt[3]{27} = \sqrt[3]{ {3}^{3} } = 3 [/tex]
[tex]6) \sqrt[6]{27 \times 125} = \sqrt[6]{3375} =
\sqrt[6]{ {15}^{3} } = \sqrt{ |15| } = \sqrt{15} [/tex]
[tex]7) \sqrt[3]{ \frac{108}{32} } = \sqrt[3]{ \frac{27}{8} } = \frac{ \sqrt[3]{27} }{ \sqrt[3]{8} } = \frac{ \sqrt[3]{ {3}^{3} } }{ \sqrt[3]{ {2}^{3} } } = \frac{3}{2} [/tex]
[tex]8) \sqrt[3]{ \frac{ {a}^{6} \times {b}^{9} }{125} } = \frac{ \sqrt[3]{ {a}^{6} {b}^{9} } }{ \sqrt[3]{125} } = \frac{ \sqrt[3]{ {a}^{6} \sqrt[3]{ {b}^{9} } } }{ \sqrt[3]{ {5}^{3} } } = \frac{ {a}^{2} {b}^{3} }{5} [/tex]
