Answer:
[tex]\sqrt[6]{2}[/tex]
Step-by-step explanation:
write the equivalent expression for [tex]\frac{\sqrt{2}}{\sqrt[3]{2}}[/tex]
to simplify it we need to rationalize the denominator
we multiply top and bottom by [tex]\sqrt[3]{4}[/tex]
[tex]\frac{\sqrt{2}*\sqrt[3]{4}}{\sqrt[3]{2}*\sqrt[3]{4}}[/tex]
[tex]\frac{\sqrt{2}*\sqrt[3]{4}}{\sqrt[3]{8}}[/tex]
[tex]\frac{\sqrt{2}*\sqrt[3]{4}}{2}[/tex]
square root can be written as 1/2 and then cube root can be written as 1/3
[tex]\sqrt[3]{4} =2^\frac{2}{3}[/tex]
[tex]\frac{\sqrt{2}*\sqrt[3]{4}}{2}[/tex]
[tex]\frac{2^\frac{1}{2}*2^\frac{2}{3}}{2}[/tex]
Now add the fractions [tex]\frac{1}{2} + \frac{2}{3} =\frac{3}{6} + \frac{4}{6}=\frac{7}{6}[/tex]
[tex]\frac{2^\frac{7}{6}}{2}[/tex]
[tex]\frac{\sqrt[6]{2^7}}{2}[/tex]
[tex]\frac{2\sqrt[6]{2}}{2}[/tex]
cancel out 2 at the top and bottom
[tex]\sqrt[6]{2}[/tex]
