Express in simplest radical form with a rational denominator. (5+sqrt (7))/(2+sqrt (7)) Answer Attemptiout of 2 square

To express the given expression in simplest radical form with a rational denominator, we need to rationalize the denominator. <br /><br />Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $2+\sqrt{7}$ is $2-\sqrt{7}$.<br /><br />$\frac{5+\sqrt{7}}{2+\sqrt{7}} \times \frac{2-\sqrt{7}}{2-\sqrt{7}}$<br /><br />Step 2: Simplify the numerator and denominator separately.<br /><br />Numerator: $(5+\sqrt{7})(2-\sqrt{7}) = 10 - 5\sqrt{7} + 2\sqrt{7} - 7 = 3 - 3\sqrt{7}$<br /><br />Denominator: $(2+\sqrt{7})(2-\sqrt{7}) = 4 - 7 = -3$<br /><br />Step 3: Combine the numerator and denominator.<br /><br />$\frac{3 - 3\sqrt{7}}{-3} = -1 + \sqrt{7}$<br /><br />Therefore, the expression $\frac{5+\sqrt{7}}{2+\sqrt{7}}$ can be simplified to $-1 + \sqrt{7}$.