Which expression is equivalenit to $$ 3^{\frac{9}{2}} $$ ? $$ \frac{(\sqrt{3})^{9}}{\sqrt{3}} \frac{\sqrt[9]{3}}{2} $$

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Calculate the value by following steps:
- step0: Calculate:
  $$3^{\frac{9}{2}}$$
- step1: Transform the expression:
  $$\sqrt{3^{9}}$$
- step2: Rewrite the expression:
  $$\sqrt{3^{8}\times 3}$$
- step3: Rewrite the expression:
  $$\sqrt{3^{8}}\times \sqrt{3}$$
- step4: Rewrite the expression:
  $$3^{4}\sqrt{3}$$
- step5: Calculate:
  $$81\sqrt{3}$$
Calculate or simplify the expression $$ (\sqrt(3))^9/\sqrt(3) * \sqrt[9](3)/2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\frac{\left(\sqrt{3}\right)^{9}}{\sqrt{3}}\times \sqrt[9]{3}}{2}$$
- step1: Multiply by $$a^{-n}:$$
  $$\frac{\left(\sqrt{3}\right)^{9}\left(\sqrt{3}\right)^{-1}\sqrt[9]{3}}{2}$$
- step2: Multiply:
  $$\frac{\left(\sqrt{3}\right)^{8}\sqrt[9]{3}}{2}$$
- step3: Evaluate the power:
  $$\frac{81\sqrt[9]{3}}{2}$$
To determine which expression is equivalent to $$ 3^{\frac{9}{2}} $$, let's simplify both expressions step by step.

1. **Simplifying $$ 3^{\frac{9}{2}} $$**:
   $$
   3^{\frac{9}{2}} = 3^{4.5} = 3^4 \cdot 3^{0.5} = 81 \cdot \sqrt{3} = 81\sqrt{3}
   $$

2. **Simplifying the expression $$ \frac{(\sqrt{3})^{9}}{\sqrt{3}} \cdot \frac{\sqrt[9]{3}}{2} $$**:
   - First, simplify $$ \frac{(\sqrt{3})^{9}}{\sqrt{3}} $$:
     $$
     \frac{(\sqrt{3})^{9}}{\sqrt{3}} = \frac{3^{\frac{9}{2}}}{3^{\frac{1}{2}}} = 3^{\frac{9}{2} - \frac{1}{2}} = 3^{\frac{8}{2}} = 3^4 = 81
     $$
   - Now, simplify $$ \frac{\sqrt[9]{3}}{2} $$:
     $$
     \frac{\sqrt[9]{3}}{2} = \frac{3^{\frac{1}{9}}}{2}
     $$
   - Combine the two parts:
     $$
     81 \cdot \frac{3^{\frac{1}{9}}}{2} = \frac{81 \cdot 3^{\frac{1}{9}}}{2}
     $$

Thus, the simplified form of the second expression is:
$$
\frac{81\sqrt[9]{3}}{2}
$$

### Conclusion:
- The first expression simplifies to $$ 81\sqrt{3} $$.
- The second expression simplifies to $$ \frac{81\sqrt[9]{3}}{2} $$.

Since $$ 81\sqrt{3} $$ and $$ \frac{81\sqrt[9]{3}}{2} $$ are not equal, neither expression is equivalent to $$ 3^{\frac{9}{2}} $$.
Neither expression is equivalent to $$ 3^{\frac{9}{2}} $$.

Which expression is equivalenit to ?

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