Match the expression x^(4)/(9) with the equivalent expression from the choices given below. sqrt [9](x^4) (sqrt (x^4))^9 sqrt [4](x^9) (sqrt (x^9))^4 Choose the correct answer below. (sqrt (x^4))^9 sqrt [4](x^9) (sqrt (x^9))^4

The expression $x^{\frac{m}{n}}$ is equivalent to $\sqrt[n]{x^m}$, which can also be written as $(\sqrt[n]{x})^m$.<br /><br />In our case, $x^{\frac{4}{9}}$, we have $m=4$ and $n=9$.<br />Therefore, $x^{\frac{4}{9}} = \sqrt[9]{x^4} = (\sqrt[9]{x})^4$.<br /><br />Comparing this with the given options:<br />1. $\sqrt[9]{x^4}$ matches our derived equivalent expression.<br />2. $(\sqrt{x^4})^9$ can be written as $(x^{\frac{4}{2}})^9 = x^{18}$, which is not equivalent to $x^{\frac{4}{9}}$.<br />3. $\sqrt[4]{x^9}$ can be written as $x^{\frac{9}{4}}$, which is not equivalent to $x^{\frac{4}{9}}$.<br />4. $(\sqrt{x^9})^4$ can be written as $(x^{\frac{9}{2}})^4 = x^{18}$, which is not equivalent to $x^{\frac{4}{9}}$.<br /><br />Thus, the correct equivalent expression is $\sqrt[9]{x^4}$.<br /><br />Final Answer: The final answer is $\boxed{\sqrt [9]{x^{4}}}$