To find an expression equivalent to [tex]\(\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}\)[/tex], we will follow a step-by-step process:
1. Combine the exponents inside the parentheses:
When you multiply exponents with the same base, you add the exponents. Therefore,
[tex]\[
x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} = x^{\left(\frac{4}{3} + \frac{2}{3}\right)}
\][/tex]
Adding the exponents:
[tex]\[
\frac{4}{3} + \frac{2}{3} = \frac{6}{3} = 2
\][/tex]
So,
[tex]\[
x^{\frac{4}{3}} \cdot x^{\frac{2}{3}} = x^2
\][/tex]
2. Apply the external exponent:
The expression now is [tex]\(\left(x^2\right)^{\frac{1}{3}}\)[/tex]. When you raise a power to another power, you multiply the exponents. Therefore,
[tex]\[
\left(x^2\right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}} = x^{\frac{2}{3}}
\][/tex]
Thus, the expression equivalent to [tex]\(\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}\)[/tex] is [tex]\(\boxed{x^{\frac{2}{3}}}\)[/tex].
