1. **State the problem:** Simplify the expression $$\frac{\sqrt{7 + \sqrt{7} + h}}{\sqrt{7 - \sqrt{7} + h}}$$ where $h$ is a variable. 2. **Recall the formula and rules:** To simplify a fraction involving square roots, we often multiply numerator and denominator by the conjugate of the denominator to rationalize it. The conjugate of $\sqrt{7 - \sqrt{7} + h}$ is $\sqrt{7 + \sqrt{7} + h}$. 3. **Multiply numerator and denominator by the conjugate:** $$\frac{\sqrt{7 + \sqrt{7} + h}}{\sqrt{7 - \sqrt{7} + h}} \times \frac{\sqrt{7 + \sqrt{7} + h}}{\sqrt{7 + \sqrt{7} + h}} = \frac{7 + \sqrt{7} + h}{\sqrt{(7 - \sqrt{7} + h)(7 + \sqrt{7} + h)}}$$ 4. **Simplify the denominator using difference of squares:** $$ (7 - \sqrt{7} + h)(7 + \sqrt{7} + h) = (7 + h)^2 - (\sqrt{7})^2 = (7 + h)^2 - 7 $$ 5. **Expand and simplify:** $$ (7 + h)^2 - 7 = (7 + h)(7 + h) - 7 = (49 + 14h + h^2) - 7 = 42 + 14h + h^2 $$ 6. **Rewrite the expression:** $$ \frac{7 + \sqrt{7} + h}{\sqrt{42 + 14h + h^2}} $$ 7. **Final simplified form:** The expression simplifies to $$\frac{7 + \sqrt{7} + h}{\sqrt{h^2 + 14h + 42}}$$. This is the simplest form unless further information about $h$ is given.