Rationalize the denominator and simplify. \[ \frac{\sqrt{7}}{14-\sqrt{7}} \]

To rationalize the denominator and simplify the expression \(\frac{\sqrt{7}}{14 - \sqrt{7}}\), follow these steps: 1. **Multiply by the Conjugate:** Multiply both the numerator and the denominator by the conjugate of the denominator, which is \(14 + \sqrt{7}\): \[ \frac{\sqrt{7}}{14 - \sqrt{7}} \times \frac{14 + \sqrt{7}}{14 + \sqrt{7}} = \frac{\sqrt{7}(14 + \sqrt{7})}{(14 - \sqrt{7})(14 + \sqrt{7})} \] 2. **Simplify the Denominator:** Use the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \): \[ (14 - \sqrt{7})(14 + \sqrt{7}) = 14^2 - (\sqrt{7})^2 = 196 - 7 = 189 \] 3. **Expand the Numerator:** Distribute \(\sqrt{7}\): \[ \sqrt{7}(14 + \sqrt{7}) = 14\sqrt{7} + 7 \] 4. **Combine the Results:** Now, the expression looks like: \[ \frac{14\sqrt{7} + 7}{189} \] 5. **Simplify the Fraction:** Factor out a 7 from the numerator: \[ \frac{7(2\sqrt{7} + 1)}{189} = \frac{2\sqrt{7} + 1}{27} \] So, the simplified form of the original expression is: \[ \frac{2\sqrt{7} + 1}{27} \] **Answer:** After simplifying, the expression is (2 √7 + 1) divided by 27. Thus, (2√7 + 1)/27