Option B:
[tex]9^{\frac{1}{8}x}[/tex] is the equivalent expression to the given expression.
Solution:
Given expression:
[tex]$\sqrt[4]{9} ^{\frac{1}{2}x}[/tex]
To find the equivalent expression to the given expression.
[tex]$\sqrt[4]{9} ^{\frac{1}{2}x}[/tex]
Using radical rule: [tex]$\sqrt[n]{a}=a^{\frac{1}{n}}[/tex]
So that [tex]$\sqrt[4]{9} = 9^{\frac{1}{4} }[/tex].
[tex]$\sqrt[4]{9} ^{\frac{1}{2}x}=\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x}[/tex]
Using exponent rule: [tex]\left(a^{b}\right)^{c}=a^{b c}[/tex]
[tex]=9^{\frac{1}{4} \cdot \frac{1}{2} x}[/tex]
[tex]=9^{\frac{1}{8}x}[/tex]
[tex]$\sqrt[4]{9} ^{\frac{1}{2}x}=9^{\frac{1}{8}x}[/tex]
Hence [tex]9^{\frac{1}{8}x}[/tex] is the equivalent expression to the given expression.
Option B is the correct answer.