Step-by-step explanation:
[tex](1) \: \: \sqrt[3]{5} \times \sqrt[3]{ \frac{2}{5} } \times \frac{ \sqrt[3]{64} }{ \sqrt[3]{3} } \times \frac{ \sqrt[6]{9} }{ \sqrt[3]{2} } [/tex]
Note that the 1st two factors can be combined:
[tex] \sqrt[3]{5} \times \sqrt[3]{ \frac{2}{5} } = \sqrt[3]{5 \times \frac{2}{5} } = \sqrt[3]{2} [/tex]
We also know that the numerator in the 3rd factor can be rewritten as
[tex] \sqrt[3]{64} = 4[/tex]
And the numerator in the 4th term can be rewritten as
[tex] \sqrt[6]{9} =({( {3})^{2} })^{ \frac{1}{6} } = \sqrt[3]{3} [/tex]
So let's rewrite expression #1
[tex] \sqrt[3]{2} \times \frac{4}{ \sqrt[3]{3} } \times \frac{ \sqrt[3]{3} }{ \sqrt[3]{2} } [/tex]
Notice that all the radical terms cancel out except for 4 therefore,
[tex]\sqrt[3]{5} \times \sqrt[3]{ \frac{2}{5} } \times \frac{ \sqrt[3]{64} }{ \sqrt[3]{3} } \times \frac{ \sqrt[6]{9} }{ \sqrt[3]{2} } = 4[/tex]
