Sure, let's analyze the expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] step by step.
1. Rewrite the expression using exponents:
The expression [tex]\(\sqrt[4]{9}\)[/tex] can be written as [tex]\(9^{\frac{1}{4}}\)[/tex]. So the entire expression becomes:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x}
\][/tex]
2. Apply the power of a power rule:
The power of a power rule states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Using this rule, we can simplify:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} = 9^{\frac{1}{4} \cdot \frac{1}{2} x}
\][/tex]
3. Simplify the exponent:
Multiply the exponents together:
[tex]\[
\frac{1}{4} \cdot \frac{1}{2} x = \frac{1}{8} x
\][/tex]
4. Rewrite the simplified expression:
Now we have:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
So the expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] is equivalent to [tex]\(9^{\frac{1}{8} x}\)[/tex].
Therefore, the correct option is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
Hence, the second option is the equivalent expression.
