Answer:
[tex]\huge\boxed{\sqrt[9]{3}}[/tex]
Step-by-step explanation:
In order to convert the expression [tex](3^{\frac{2}{3}})^{\frac{1}{6}}[/tex] into a radical, we need to simplify it down first.
Using exponent rules, we know that [tex](a^b)^c[/tex] can be simplified to [tex]a^{b\cdot c}[/tex].
Therefore we can say that [tex](3^{\frac{2}{3}})^{\frac{1}{6}}[/tex] is the same as [tex]3^{\frac{2}{3} \cdot \frac{1}{6}}[/tex].
[tex]\frac{2}{3} \cdot \frac{1}{6} = \frac{2}{18} = \frac{1}{9}[/tex]
So we have [tex]3^{\frac{1}{9}}[/tex].
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 [tex]2^{\frac{3}{4}}[/tex]:
We'll take the denominator root (the denominator is 4) of the base (2):
[tex]\sqrt[4]{2}[/tex]
But inside the radical, we raise the base to the numerator (3) power.
[tex]\sqrt[4]{2^3}[/tex]
Same logic for [tex]3^{\frac{1}{9}}[/tex]
[tex]\sqrt[9]{3^1}[/tex]
[tex]3^1[/tex] is the same as 3.
[tex]\sqrt[9]{3}[/tex]
Hope this helped!
