Question

Square root of 2 divided by the cube root of 2

Answer 1

`\frac{ √(2) }{ \sqrt[3]{2} } = \sqrt[6]{2}`

EXPLANATION

We want to simplify

`\frac{ √(2) }{ \sqrt[3]{2} }`

We rewrite in exponential form to get:

`\frac{ √(2) }{ \sqrt[3]{2} } = \frac{ {2}^{ (1)/(2) } }{{2}^{ (1)/(3) } }`

Recall the quotient rule of exponents:

`\frac{ {a}^(m) }{ {a}^(n) } = {a}^(m - n)`

We apply this rule to get:

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

Simplify:

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

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

`\frac{ √(2) }{ \sqrt[3]{2} } = \sqrt[6]{2}`

Answer 2

Answer:
`\sqrt[6]{2}`

Explanation:

You know that the expression is
`\frac{√(2)}{\sqrt[3]{2}}`

By definition we know that:

`\sqrt[n]{a}=a^{(1)/(n)`

You also need to remember the Quotient of powers property:

`(a^n)/(a^m)=a^((n-m))`

Therefore, you can rewrite the expression:

`=\frac{2^{(1)/(2)}}{2^{(1)/(3)}}`

Finally, you have to simplify the expression. Therefore, you get:

`=2^{((1)/(2)-(1)/(3))}\\=2^{(1)/(6)}\\=\sqrt[6]{2}`