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

To find an expression equivalent to \( 2 y^{\frac{7}{2}} \), let's rewrite it in a simpler form. The term \( y^{\frac{7}{2}} \) can be expressed as \( 2 \sqrt{y^7} \) since \( y^{\frac{7}{2}} = \sqrt{y^7} \). Thus, \( 2 y^{\frac{7}{2}} = 2 \sqrt{y^7} \). Now, looking at the options provided, the first choice \( 2 \sqrt{y^{7}} \) is indeed equivalent to \( 2 y^{\frac{7}{2}} \). For the second option, simplifying \( \sqrt{(2 y)^{7}} \frac{2 \sqrt[7]{y}}{\sqrt{y}} \) does not yield the same result, since the multiplication and exponents involved would change the initial expression. So the correct answer is \( 2 \sqrt{y^{7}} \). In general, understanding exponents is essential for algebra, as it lets you transform expressions neatly without a calculator—perfect for impressing your friends with your math skills! Plus, knowing these transformations can speed up problem-solving in intricate equations.