Question

Rewrite the expression with a rational exponent as a radical expression.

three to the two thirds power all raised to the one sixth power

Answer 1

`\huge\boxed{\sqrt[9]{3}}`

Explanation:

In order to convert the expression
`(3^{(2)/(3)})^{(1)/(6)}` into a radical, we need to simplify it down first.

Using exponent rules, we know that
`(a^b)^c` can be simplified to
`a^(b\cdot c)`.

Therefore we can say that
`(3^{(2)/(3)})^{(1)/(6)}` is the same as
`3^{(2)/(3) \cdot (1)/(6)}`.

`(2)/(3) \cdot (1)/(6) = (2)/(18) = (1)/(9)`

So we have
`3^{(1)/(9)}`.

When we have a number to a fraction power, it's the same thing as taking the denominator root of the base to the numerator power.

Basically:

If we have
`2^{(3)/(4)}`:

We'll take the denominator root (the denominator is 4) of the base (2):

`\sqrt[4]{2}`

But inside the radical, we raise the base to the numerator (3) power.

`\sqrt[4]{2^3}`

Same logic for
`3^{(1)/(9)}`

`\sqrt[9]{3^1}`

`3^1` is the same as 3.

`\sqrt[9]{3}`

Hope this helped!