To find an equivalent expression for [tex]\(\sqrt[4]{9^{\frac{1}{2}} x}\)[/tex], we need to simplify it step by step.
First, consider the expression:
[tex]\[\sqrt[4]{9^{\frac{1}{2}} x}.\][/tex]
We'll start by simplifying the term inside the fourth root.
[tex]\(9^{\frac{1}{2}}\)[/tex] means "9 to the power of one-half", which is the square root of 9:
[tex]\[9^{\frac{1}{2}} = \sqrt{9} = 3.\][/tex]
So the expression inside the fourth root becomes:
[tex]\[ \sqrt[4]{3 \cdot x}. \][/tex]
Next, we will work on the fourth root:
[tex]\[ \sqrt[4]{3 \cdot x} = (3 \cdot x)^{\frac{1}{4}}. \][/tex]
Now, split the power across the product:
[tex]\[ (3 \cdot x)^{\frac{1}{4}} = 3^{\frac{1}{4}} \cdot x^{\frac{1}{4}}. \][/tex]
Now, we need to express the [tex]\(3^{\frac{1}{4}}\)[/tex] term in a base of 9 if possible, since the given options are generally in terms of base 9.
Recall that:
[tex]\[ 3 = 9^{\frac{1}{2}}. \][/tex]
So:
[tex]\[ 3^{\frac{1}{4}} = \left(9^{\frac{1}{2}}\right)^{\frac{1}{4}}. \][/tex]
Using the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ \left(9^{\frac{1}{2}}\right)^{\frac{1}{4}} = 9^{\frac{1}{2} \cdot \frac{1}{4}} = 9^{\frac{1}{8}}. \][/tex]
Thus:
[tex]\[ 3^{\frac{1}{4}} = 9^{\frac{1}{8}}. \][/tex]
Therefore, the entire expression can now be written as:
[tex]\[ 3^{\frac{1}{4}} \cdot x^{\frac{1}{4}} = 9^{\frac{1}{8}} \cdot x^{\frac{1}{4}}. \][/tex]
None of the given options contain [tex]\(9^{2x}\)[/tex] or [tex]\(\sqrt[5]{9}^x\)[/tex]. The correct answer based on the simplification is:
[tex]\[ 9^{\frac{1}{8} x}. \][/tex]
Therefore, the expression [tex]\(\sqrt[4]{9^{\frac{1}{2}} x}\)[/tex] is equivalent to:
\[ \boxed{9^{\frac{1}{8} x}}.
