To solve the expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex], let's break it down step by step to find an equivalent expression.
1. Rewrite the nth-root as a fractional exponent:
[tex]\[
\sqrt[4]{9} = 9^{\frac{1}{4}}
\][/tex]
So, the given expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] can be rewritten using this equivalent form:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x}
\][/tex]
2. Apply the power rule for exponents:
The power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] allows us to simplify the nested exponent:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} = 9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)}
\][/tex]
3. Simplify the exponent:
Multiply the exponents inside the power:
[tex]\[
9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)} = 9^{\frac{1}{8} x}
\][/tex]
Therefore, the expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] is equivalent to [tex]\(9^{\frac{1}{8} x}\)[/tex].
By comparing this result with the given options:
- [tex]\(9^{2 x}\)[/tex]
- [tex]\(9^{\frac{1}{8} x}\)[/tex]
- [tex]\(\sqrt{9}^x\)[/tex]
- [tex]\(\sqrt[6]{9^x}\)[/tex]
The correct equivalent expression is:
[tex]\[
\boxed{9^{\frac{1}{8} x}}
\][/tex]
