Rationalize the denominator. \( \frac{7}{1+\sqrt{7}} \) (Simplify your answer. Type an exact answer, using radicals as needed.) (iver

To rationalize the denominator of \( \frac{7}{1+\sqrt{7}} \), multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1-\sqrt{7} \): \[ \frac{7(1-\sqrt{7})}{(1+\sqrt{7})(1-\sqrt{7})} \] Now, the denominator simplifies using the difference of squares: \[ (1+\sqrt{7})(1-\sqrt{7}) = 1^2 - (\sqrt{7})^2 = 1 - 7 = -6 \] Thus, the expression becomes: \[ \frac{7(1-\sqrt{7})}{-6} \] Now simplify the numerator: \[ = \frac{7 - 7\sqrt{7}}{-6} = \frac{-7 + 7\sqrt{7}}{6} \] So, the rationalized form of the expression is: \[ \frac{7\sqrt{7} - 7}{6} \]