To determine which of the given options is equivalent to [tex]\(9^{\frac{3}{4}}\)[/tex], let's analyze each one step-by-step.
1. Given Expression: [tex]\(9^{\frac{3}{4}}\)[/tex]
- [tex]\( 9 \)[/tex] can be expressed as [tex]\( 3^2 \)[/tex]:
- Therefore, [tex]\( 9^{\frac{3}{4}} = (3^2)^{\frac{3}{4}} \)[/tex].
- By the properties of exponents, we know [tex]\((a^m)^n = a^{mn}\)[/tex]:
- So, [tex]\((3^2)^{\frac{3}{4}} = 3^{2 \cdot \frac{3}{4}} = 3^{\frac{6}{4}} = 3^{\frac{3}{2}}\)[/tex].
2. Evaluate Each Option:
- Option A: [tex]\(\sqrt[3]{9}\)[/tex]
- Using the properties of exponents, [tex]\(\sqrt[3]{9}\)[/tex] can be written as [tex]\(9^{\frac{1}{3}}\)[/tex].
- Expressing 9 as [tex]\(3^2\)[/tex]: [tex]\(\sqrt[3]{9} = (3^2)^{\frac{1}{3}} = 3^{2 \cdot \frac{1}{3}} = 3^{\frac{2}{3}}\)[/tex].
- This does not match [tex]\(3^{\frac{3}{2}}\)[/tex].
- Option B: [tex]\(\sqrt[4]{9}\)[/tex]
- Similarly, [tex]\(\sqrt[4]{9} = 9^{\frac{1}{4}}\)[/tex].
- Expressing 9 as [tex]\(3^2\)[/tex]: [tex]\(\sqrt[4]{9} = (3^2)^{\frac{1}{4}} = 3^{2 \cdot \frac{1}{4}} = 3^{\frac{1}{2}}\)[/tex].
- This does not match [tex]\(3^{\frac{3}{2}}\)[/tex].
- Option C: [tex]\(\sqrt{3}\)[/tex]
- This can be rewritten as [tex]\(3^{\frac{1}{2}}\)[/tex].
- Clearly, [tex]\(3^{\frac{1}{2}}\)[/tex] is not equal to [tex]\(3^{\frac{3}{2}}\)[/tex].
- Option D: [tex]\(3 \sqrt{3}\)[/tex]
- Express [tex]\(3\)[/tex] as [tex]\(3^1\)[/tex] and [tex]\(\sqrt{3}\)[/tex] as [tex]\(3^{\frac{1}{2}}\)[/tex].
- Therefore, [tex]\(3 \sqrt{3} = 3^1 \cdot 3^{\frac{1}{2}} = 3^{1 + \frac{1}{2}} = 3^{\frac{3}{2}}\)[/tex].
- This matches [tex]\(9^{\frac{3}{4}} = 3^{\frac{3}{2}}\)[/tex].
Thus, the expression [tex]\(9^{\frac{3}{4}}\)[/tex] is equivalent to Option D: [tex]\(3 \sqrt{3}\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{4}\)[/tex].
