Which expression is equivalent to $$ 2 y^{\frac{7}{2}} $$ ? $$ 2 \sqrt{y^{7}} 2 \sqrt[7]{y^{2}} $$ $$ \sqrt{(2 y)^{7}} \frac{2 \sqrt[7]{y}}{\sqrt{y}} $$

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To find which expression is equivalent to $$ 2 y^{\frac{7}{2}} $$, we need to simplify each of the given expressions and compare them with the original expression.

1. $$ 2 \sqrt{y^{7}} $$:
   - Simplify $$ \sqrt{y^{7}} $$ as $$ y^{\frac{7}{2}} $$.
   - Multiply by 2 to get $$ 2y^{\frac{7}{2}} $$.
   - This expression is equivalent to $$ 2 y^{\frac{7}{2}} $$.

2. $$ 2 \sqrt[7]{y^{2}} $$:
   - Simplify $$ \sqrt[7]{y^{2}} $$ as $$ y^{\frac{2}{7}} $$.
   - Multiply by 2 to get $$ 2y^{\frac{2}{7}} $$.
   - This expression is not equivalent to $$ 2 y^{\frac{7}{2}} $$.

3. $$ \sqrt{(2 y)^{7}} $$:
   - Simplify $$ (2 y)^{7} $$ as $$ 2^{7}y^{7} $$.
   - Take the square root to get $$ 2^{3}y^{\frac{7}{2}} $$.
   - This expression is not equivalent to $$ 2 y^{\frac{7}{2}} $$.

4. $$ \frac{2 \sqrt[7]{y}}{\sqrt{y}} $$:
   - Simplify $$ \sqrt[7]{y} $$ as $$ y^{\frac{1}{7}} $$ and $$ \sqrt{y} $$ as $$ y^{\frac{1}{2}} $$.
   - Multiply $$ y^{\frac{1}{7}} $$ by $$ y^{\frac{1}{2}} $$ to get $$ y^{\frac{1}{7}+\frac{1}{2}} = y^{\frac{9}{14}} $$.
   - Multiply by 2 to get $$ 2y^{\frac{9}{14}} $$.
   - This expression is not equivalent to $$ 2 y^{\frac{7}{2}} $$.

Therefore, the expression that is equivalent to $$ 2 y^{\frac{7}{2}} $$ is $$ 2 \sqrt{y^{7}} $$.
The equivalent expression is $$ 2 \sqrt{y^{7}} $$.

Which expression is equivalent to ?

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