Question

Which expression is equivalent to sqrt 2 / ^3 sqrt 2?

Answer 1

`\displaystyle\\ √(2)=2^{^(1)/(2)}\\\\\sqrt[3]{2}=2^{^(1)/(3)}\\\\\frac{√(2)}{\sqrt[3]{2}}=\frac{2^{^(1)/(2)}}{2^{^(1)/(3)}}=2^{^{(1)/(2)-(1)/(3)}}=2^{^{(3-2)/(6)}}=2^{^{(1)/(6)}}=\boxed{\sqrt[\b6]{2}}`

Answer 2

`\sqrt[6]{2}`

Explanation:

write the equivalent expression for
`\frac{√(2)}{\sqrt[3]{2}}`

to simplify it we need to rationalize the denominator

we multiply top and bottom by
`\sqrt[3]{4}`

`\frac{√(2)*\sqrt[3]{4}}{\sqrt[3]{2}*\sqrt[3]{4}}`

`\frac{√(2)*\sqrt[3]{4}}{\sqrt[3]{8}}`

`\frac{√(2)*\sqrt[3]{4}}{2}`

square root can be written as 1/2 and then cube root can be written as 1/3

`\sqrt[3]{4} =2^(2)/(3)`

`\frac{√(2)*\sqrt[3]{4}}{2}`

`(2^(1)/(2)*2^(2)/(3))/(2)`

Now add the fractions
`(1)/(2) + (2)/(3) =(3)/(6) + (4)/(6)=(7)/(6)`

`(2^(7)/(6))/(2)`

`\frac{\sqrt[6]{2^7}}{2}`

`\frac{2\sqrt[6]{2}}{2}`

cancel out 2 at the top and bottom

`\sqrt[6]{2}`