Rationalize the denominator. (y)/(sqrt (7)+sqrt (y)) square

To rationalize the denominator of the expression \(\frac{y}{\sqrt{7} + \sqrt{y}}\), we need to eliminate the square roots in the denominator. We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{7} + \sqrt{y}\) is \(\sqrt{7} - \sqrt{y}\).<br /><br />Here are the steps:<br /><br />1. Multiply the numerator and the denominator by the conjugate of the denominator:<br /> \[<br /> \frac{y}{\sqrt{7} + \sqrt{y}} \times \frac{\sqrt{7} - \sqrt{y}}{\sqrt{7} - \sqrt{y}}<br /> \]<br /><br />2. Apply the distributive property (FOIL method) to both the numerator and the denominator:<br /> \[<br /> \frac{y (\sqrt{7} - \sqrt{y})}{(\sqrt{7} + \sqrt{y})(\sqrt{7} - \sqrt{y})}<br /> \]<br /><br />3. Simplify the denominator using the difference of squares formula \((a + b)(a - b) = a^2 - b^2\):<br /> \[<br /> (\sqrt{7} + \sqrt{y})(\sqrt{7} - \sqrt{y}) = (\sqrt{7})^2 - (\sqrt{y})^2 = 7 - y<br /> \]<br /><br />4. Substitute back into the fraction:<br /> \[<br /> \frac{y (\sqrt{7} - \sqrt{y})}{7 - y}<br /> \]<br /><br />Thus, the rationalized form of the given expression is:<br />\[<br />\boxed{\frac{y (\sqrt{7} - \sqrt{y})}{7 - y}}<br />\]