1. **Problem Statement:** Simplify the complex rational expression. For example, simplify $$\frac{\frac{a}{b} + \frac{c}{d}}{\frac{e}{f} - \frac{g}{h}}$$. 2. **Formula and Rules:** To simplify complex rational expressions, first find a common denominator for the numerator and denominator separately, then combine the fractions. Finally, divide the numerator by the denominator by multiplying by the reciprocal. 3. **Step-by-step Simplification:** - Find common denominators: $$\frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{cb}{bd} = \frac{ad + cb}{bd}$$ $$\frac{e}{f} - \frac{g}{h} = \frac{eh}{fh} - \frac{gf}{fh} = \frac{eh - gf}{fh}$$ - Rewrite the complex fraction: $$\frac{\frac{ad + cb}{bd}}{\frac{eh - gf}{fh}}$$ - Multiply numerator by reciprocal of denominator: $$\frac{ad + cb}{bd} \times \frac{fh}{eh - gf}$$ - Multiply numerators and denominators: $$\frac{(ad + cb) \times fh}{bd \times (eh - gf)}$$ 4. **Explanation:** We combined the fractions in numerator and denominator by finding common denominators, then converted the complex fraction into a simple fraction by multiplying by the reciprocal of the denominator. This method works for any complex rational expression. 5. **Final Answer:** $$\frac{(ad + cb)fh}{bd(eh - gf)}$$