To determine which expression is equivalent to [tex]\(\left( x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} \right)^{\frac{1}{3}}\)[/tex], follow these steps:
1. Combine the exponents of [tex]\(x\)[/tex] inside the parentheses:
According to the properties of exponents, when you multiply like bases, you add the exponents. Therefore:
[tex]\[
x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{4}{3} + \frac{2}{3}}
\][/tex]
2. Add the exponents:
[tex]\[
\frac{4}{3} + \frac{2}{3} = \frac{4 + 2}{3} = \frac{6}{3} = 2
\][/tex]
So the expression simplifies to:
[tex]\[
\left( x^2 \right)^{\frac{1}{3}}
\][/tex]
3. Apply the power of a power property:
According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Therefore:
[tex]\[
\left( x^2 \right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} = x^{\frac{2}{3}}
\][/tex]
Therefore, the expression that is equivalent to [tex]\(\left( x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} \right)^{\frac{1}{3}}\)[/tex] is:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{x^{\frac{2}{3}}}
\][/tex]
