To determine [tex]\(6^{\frac{2}{3}}\)[/tex] in radical form, we need to convert the expression involving the fractional exponent into an expression involving a radical (root).
The fractional exponent [tex]\(\frac{2}{3}\)[/tex] can be interpreted as follows:
[tex]\[6^{\frac{2}{3}} = (6^2)^{\frac{1}{3}}\][/tex]
This is because:
[tex]\[a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}}\][/tex]
Here, we have [tex]\(a = 6\)[/tex], [tex]\(m = 2\)[/tex], and [tex]\(n = 3\)[/tex]:
[tex]\[6^{\frac{2}{3}} = (6^2)^{\frac{1}{3}}\][/tex]
Now, let's calculate [tex]\(6^2\)[/tex]:
[tex]\[6^2 = 36\][/tex]
Therefore, we can express [tex]\(6^{\frac{2}{3}}\)[/tex] as:
[tex]\[(6^2)^{\frac{1}{3}} = \sqrt[3]{6^2} = \sqrt[3]{36}\][/tex]
Among the given options:
- [tex]\(\sqrt[3]{6^2}\)[/tex] is the correct radical form.
- [tex]\(\sqrt[2]{6^3}\)[/tex] would be [tex]\(6^{\frac{3}{2}}\)[/tex].
- [tex]\(\sqrt[2]{6 \cdot 3}\)[/tex] would be [tex]\(\sqrt{18}\)[/tex].
So, the correct radical form for [tex]\(6^{\frac{2}{3}}\)[/tex] is:
[tex]\[\boxed{\sqrt[3]{6^2}}\][/tex]
When evaluated, this value is approximately:
[tex]\[6^{\frac{2}{3}} \approx 3.3019272488946263\][/tex]
So the correct radical form of [tex]\(6^{\frac{2}{3}}\)[/tex] is indeed [tex]\(\sqrt[3]{6^2}\)[/tex].
