Answer:
Options (1), (2), (3) and (7)
Step-by-step explanation:
Given expression is [tex]\frac{\sqrt[3]{8^{\frac{1}{3}}\times 3} }{3\times2^{\frac{1}{9}}}[/tex].
Now we will solve this expression with the help of law of exponents.
[tex]\frac{\sqrt[3]{8^{\frac{1}{3}}\times 3} }{3\times2^{\frac{1}{9}}}=\frac{\sqrt[3]{(2^3)^{\frac{1}{3}}\times 3} }{3\times2^{\frac{1}{9}}}[/tex]
[tex]=\frac{\sqrt[3]{2\times 3} }{3\times2^{\frac{1}{9}}}[/tex]
[tex]=\frac{2^{\frac{1}{3}}\times 3^{\frac{1}{3}}}{3\times 2^{\frac{1}{9}}}[/tex]
[tex]=2^{\frac{1}{3}}\times 3^{\frac{1}{3}}\times 2^{-\frac{1}{9}}\times 3^{-1}[/tex]
[tex]=2^{\frac{1}{3}-\frac{1}{9}}\times 3^{\frac{1}{3}-1}[/tex]
[tex]=2^{\frac{3-1}{9}}\times 3^{\frac{1-3}{3}}[/tex]
[tex]=2^{\frac{2}{9}}\times 3^{-\frac{2}{3} }[/tex] [Option 2]
[tex]2^{\frac{2}{9}}\times 3^{-\frac{2}{3} }=(\sqrt[9]{2})^2\times (\sqrt[3]{\frac{1}{3} } )^2[/tex] [Option 1]
[tex]2^{\frac{2}{9}}\times 3^{-\frac{2}{3} }=(\sqrt[9]{2})^2\times (\sqrt[3]{\frac{1}{3} } )^2[/tex]
[tex]=(2^2)^{\frac{1}{9}}\times (3^2)^{-\frac{1}{3} }[/tex]
[tex]=\sqrt[9]{4}\times \sqrt[3]{\frac{1}{9} }[/tex] [Option 3]
[tex]2^{\frac{2}{9}}\times 3^{-\frac{2}{3} }=(2^2)^{\frac{1}{9}}\times (3^{-2})^{\frac{1}{3} }[/tex]
[tex]=\sqrt[9]{2^2}\times \sqrt[3]{3^{-2}}[/tex] [Option 7]
Therefore, Options (1), (2), (3) and (7) are the correct options.