Let's solve the given expression [tex]\(\sqrt[4]{9^{\frac{1}{2}} x}\)[/tex] step by step.
1. Understanding the expression:
- [tex]\(\sqrt[4]{9^{\frac{1}{2}} x}\)[/tex] can be written as [tex]\((9^{\frac{1}{2}} x)^{\frac{1}{4}}\)[/tex].
2. Decomposing the expression:
- The expression [tex]\((9^{\frac{1}{2}} x)^{\frac{1}{4}}\)[/tex] can be separated into two parts: [tex]\(9^{\frac{1}{2}}\)[/tex] and [tex]\(x\)[/tex].
- For the first part, we have [tex]\((9^{\frac{1}{2}})^{\frac{1}{4}}\)[/tex].
- For the second part, we have [tex]\(x^{\frac{1}{4}}\)[/tex].
3. Simplify the first part:
- Using the law of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify [tex]\((9^{\frac{1}{2}})^{\frac{1}{4}}\)[/tex] as:
[tex]\[
(9^{\frac{1}{2}})^{\frac{1}{4}} = 9^{\left(\frac{1}{2} \times \frac{1}{4}\right)} = 9^{\frac{1}{8}}
\][/tex]
4. Combine the simplified parts:
- Therefore, the expression [tex]\((9^{\frac{1}{2}} x)^{\frac{1}{4}}\)[/tex] becomes:
[tex]\[
9^{\frac{1}{8}} \cdot x^{\frac{1}{4}}
\][/tex]
5. Reviewing the given choices:
- [tex]\(9^{2x}\)[/tex]
- [tex]\(9^{\frac{1}{8}} \times\)[/tex]
- [tex]\(\sqrt{9}^x\)[/tex]
- [tex]\(\sqrt[5]{9} x\)[/tex]
Among these choices, the expression [tex]\( 9^{\frac{1}{8}} \)[/tex] closely matches our simplified form [tex]\(9^{\frac{1}{8}}\)[/tex].
Hence, the expression [tex]\(\sqrt[4]{9^{\frac{1}{2}} x}\)[/tex] is equivalent to:
[tex]\[ 9^{\frac{1}{8}} \times \][/tex]
