Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.

Express \(\frac{2}{3 - \sqrt{7}} \text{ in the form} a + \sqrt{b}\), where a and b are integers.

Answer Details

We can begin by multiplying both the numerator and denominator by the conjugate of the denominator, which is \(3 + \sqrt{7}\). This is done to eliminate the square root in the denominator. \[\frac{2}{3 - \sqrt{7}} \times \frac{3 + \sqrt{7}}{3 + \sqrt{7}} = \frac{2(3 + \sqrt{7})}{9 - 7} = \frac{2(3 + \sqrt{7})}{2} = 3 + \sqrt{7}\] Therefore, \(\frac{2}{3 - \sqrt{7}} = 3 + \sqrt{7}\), and the answer is (B) \(3 + \sqrt{7}\).