To simplify a complex fraction, calculate common denominators for both the numerator and the denominator, combine like terms, and then solve the complex fraction by flipping the denominator to multiply. Dividing and multiplying are inverse operations, so flipping is the easiest way to solve.
- Find common denominators for the numerator and denominator
Consider the example (6 + 1/x) / (8 + 4/x^2). Notice that the common denominator for the numerator term is x, and that the denominator's common denominator is x^2. Multiply other terms in an equivalent way to combine like terms. Write the example problem in this way: (6x/x + 1/x) / (8x^2/x^2 + 4/x^2).
- Combine like terms in the numerator and denominator
Add like terms in both the numerator and the denominator. Write the next step for the example problem as such: [(6x + 1) / x] / [(8x^2 + 4) / x^2].
- Flip the denominator, and multiply to solve the problem
Remember that the central operation in a fraction is division. Flipping the numerator and denominator within the denominator makes that the inverse, and multiplying by a term's inverse is equivalent to dividing by that term. Write the next step for the example problem in this way: [(6x + 1) / x] * [x^2 / (8x^2 + 4)]. Multiply across to yield this result: (6x+1)(x^2) / x(8x^2 + 4). Multiply out to get this result: 6x^3 + x^2 / 8x^3 + 4x. Since there are no common factors in this case, this is the final answer. Write the final solution as such: 6x^3 + x^2 / 8x^3 + 4x.