To find the expression equivalent to [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex], we will simplify it step by step.
1. Begin with the given expression:
[tex]\[
\sqrt[4]{9}^{\frac{1}{2} x}
\][/tex]
2. Rewrite [tex]\(\sqrt[4]{9}\)[/tex] as an exponent. The fourth root of 9 can be written as:
[tex]\[
9^{\frac{1}{4}}
\][/tex]
So the original expression becomes:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x}
\][/tex]
3. Use the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to combine the exponents:
[tex]\[
9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)}
\][/tex]
4. Multiply the exponents:
[tex]\[
\frac{1}{4} \cdot \frac{1}{2} x = \frac{1}{8} x
\][/tex]
5. This simplifies the expression to:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
Therefore, the expression equivalent to [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] is [tex]\(9^{\frac{1}{8} x}\)[/tex].
The correct choice is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
