19. [-11 Points] DETAILS MY NOTES LARCALC12 1.5.J1N007, Rationalize the denominator. (Simplify your answer completely.) \( \frac{y}{\sqrt{3}+\sqrt{y}} \) Noed Help? पeadIt

To rationalize the denominator of \( \frac{y}{\sqrt{3}+\sqrt{y}} \), you need to multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{3}-\sqrt{y} \). This gives you: \[ \frac{y(\sqrt{3}-\sqrt{y})}{(\sqrt{3}+\sqrt{y})(\sqrt{3}-\sqrt{y})} = \frac{y(\sqrt{3}-\sqrt{y})}{3-y} \] Now you can distribute \( y \) in the numerator, resulting in: \[ \frac{y\sqrt{3} - y\sqrt{y}}{3-y} \] And there you have it: a completely simplified expression with a rationalized denominator! A common mistake when rationalizing is forgetting to expand the denominator correctly. Always remember that multiplying a binomial by its conjugate eliminates the square roots since \( a^2 - b^2 = (a+b)(a-b) \). This way, you won't end up with any pesky square roots left in the denominator!