To express [tex]\(4 x^{\frac{2}{3}}\)[/tex] in radical form, we need to understand the relationship between exponents and radicals.
The general form to convert an exponent [tex]\(a^{\frac{m}{n}}\)[/tex] into a radical is:
[tex]\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \][/tex]
In our specific case, we have the expression [tex]\(4 x^{\frac{2}{3}}\)[/tex]. This can be broken down as follows:
1. The coefficient [tex]\(4\)[/tex] is left as it is because it is not affected by the exponent.
2. The term [tex]\(x^{\frac{2}{3}}\)[/tex] needs to be converted to radical form.
Using the rule [tex]\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)[/tex], we convert [tex]\(x^{\frac{2}{3}}\)[/tex]:
[tex]\[ x^{\frac{2}{3}} = \sqrt[3]{x^2} \][/tex]
Therefore, combining both parts (the coefficient and the radical expression for [tex]\(x^{\frac{2}{3}}\)[/tex]), we get:
[tex]\[ 4 x^{\frac{2}{3}} = 4 \sqrt[3]{x^2} \][/tex]
Thus, the correct radical form of the expression is:
[tex]\[ 4 \sqrt[3]{x^2} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{4 \sqrt[3]{x^2}} \][/tex]
