Let's simplify the given expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] step-by-step.
### Step 1: Express the Fourth Root in Exponential Form
First, we recognize that the fourth root of a number can be written as a power. The fourth root of [tex]\(9\)[/tex] is expressed as:
[tex]\[
\sqrt[4]{9} = 9^{\frac{1}{4}}
\][/tex]
### Step 2: Raise to the Power of [tex]\(\frac{1}{2} x\)[/tex]
Next, we raise [tex]\(9^{\frac{1}{4}}\)[/tex] to the power of [tex]\(\frac{1}{2} x\)[/tex]:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x}
\][/tex]
### Step 3: Use the Exponentiation Property
We apply the property of exponents that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. This allows us to combine the exponents:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} = 9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)}
\][/tex]
### Step 4: Simplify the Exponent
Now, let's multiply the exponents inside the parentheses:
[tex]\[
9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)} = 9^{\frac{1}{8} x}
\][/tex]
So, the simplified equivalent expression of [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
Therefore, among the given choices, the correct answer is:
[tex]\[
9^{\frac{1}{8} x}
\][/tex]
