To solve the expression [tex]\(9^{\frac{1}{4}}\)[/tex], we need to find the value of this fourth root.
The expression [tex]\(9^{\frac{1}{4}}\)[/tex] indicates the fourth root of 9. We want to find what this value is and which of the multiple choice options it equals.
### Step-by-Step Solution:
1. Understand the Power and Root Relationship:
- [tex]\(9^{\frac{1}{4}}\)[/tex] means [tex]\( \sqrt[4]{9} \)[/tex].
2. Simplify Under the Root:
- 9 can be expressed as [tex]\(3^2\)[/tex].
- Therefore, [tex]\(9^{\frac{1}{4}}\)[/tex] becomes [tex]\((3^2)^{\frac{1}{4}}\)[/tex].
- Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get [tex]\(3^{2 \cdot \frac{1}{4}} = 3^{\frac{1}{2}}\)[/tex].
3. Understand What [tex]\(3^{\frac{1}{2}}\)[/tex] Represents:
- [tex]\(3^{\frac{1}{2}}\)[/tex] is the square root of 3, or [tex]\(\sqrt{3}\)[/tex].
Given these calculations, the answer is:
[tex]\[
9^{\frac{1}{4}} = \sqrt{3}
\][/tex]
### Comparing to Options:
Now let's compare to the given options:
- A) [tex]\(\sqrt[3]{9}\)[/tex]: This is the cube root of 9, which does not match our result.
- B) [tex]\(\sqrt[4]{9}\)[/tex]: This is equivalent to [tex]\(9^{\frac{1}{4}}\)[/tex], but not in a simplified form.
- C) [tex]\(\sqrt{3}\)[/tex]: This matches our simplified result, as we have shown [tex]\(3^{\frac{1}{2}} = \sqrt{3}\)[/tex].
- D) [tex]\(3 \sqrt{3}\)[/tex]: This represents three times the square root of 3, which is not correct.
Therefore, the correct option is:
[tex]\[
\boxed{C}
\][/tex]
