To determine which expression is equivalent to [tex]\(\sqrt[4]{9^{\frac{1}{2} x}}\)[/tex], we need to simplify the given expression step by step using properties of exponents and radicals.
Given expression:
[tex]\[
\sqrt[4]{9^{\frac{1}{2} x}}
\][/tex]
We can rewrite the fourth root using exponent notation. The fourth root of a number is the same as raising that number to the power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\sqrt[4]{a} = a^{\frac{1}{4}}
\][/tex]
Applying this to our given expression:
[tex]\[
\sqrt[4]{9^{\frac{1}{2} x}} = \left(9^{\frac{1}{2} x}\right)^{\frac{1}{4}}
\][/tex]
Next, we use the property of exponents that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this property:
[tex]\[
\left(9^{\frac{1}{2} x}\right)^{\frac{1}{4}} = 9^{\left(\frac{1}{2} x \cdot \frac{1}{4}\right)} = 9^{\frac{1}{8} x}
\][/tex]
Thus, the simplified form of the expression [tex]\(\sqrt[4]{9^{\frac{1}{2} x}}\)[/tex] is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
Hence, the correct answer is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
So, the equivalent expression to [tex]\(\sqrt[4]{9^{\frac{1}{2} x}}\)[/tex] is:
[tex]\[
\boxed{9^{\frac{1}{8} x}}
\][/tex]
