A complex fraction has fractions inside its numerator or denominator, like $\dfrac{1 - \frac{4}{x^2}}{1 + \frac{2}{x}}$. You can untangle it with tools from Adding and Subtracting Rational Expressions and Multiplying and Dividing Rational Expressions.
Key idea
Find the LCD of all the small fractions. Multiply the top and the bottom of the big fraction by it; every small denominator cancels. Then factor and simplify. (Another way: combine the top into one fraction, the bottom into one fraction, then divide.)
Why it works
Multiplying the numerator and the denominator by the same nonzero expression is multiplying by 1, so the value doesn't change. Choosing the LCD makes every small fraction's denominator cancel at once, leaving a plain rational expression.
Worked examples
Worked example: Clear with the LCD
Simplify $\dfrac{1 - \frac{4}{x^2}}{1 + \frac{2}{x}}$.
- The small denominators are $x^2$ and $x$; the LCD is $x^2$.
- Multiply top and bottom by $x^2$: $\frac{x^2 - 4}{x^2 + 2x}$.
- Factor and cancel: $\frac{(x - 2)(x + 2)}{x(x + 2)} = \frac{x - 2}{x}$.
Answer: $\frac{x - 2}{x}$
Worked example: Combine, then divide
Simplify $\dfrac{\frac{1}{x} - \frac{1}{5}}{x - 5}$.
- Combine the top: $\frac{5 - x}{5x}$.
- Divide by $x - 5$: $\frac{5 - x}{5x(x - 5)}$.
- $5 - x = -(x - 5)$, so this is $-\frac{1}{5x}$.
Answer: $-\frac{1}{5x}$
Worked example: A difference quotient
Simplify $\dfrac{\frac{1}{x + 3} - \frac{1}{x}}{3}$.
- Combine the top over $x(x + 3)$: $\frac{x - (x + 3)}{x(x + 3)} = \frac{-3}{x(x + 3)}$.
- Divide by 3.
Answer: $-\frac{1}{x(x + 3)}$. (Expressions like this one appear in calculus.)
Common mistake
Multiplying only the small fractions by the LCD, or only the top of the big fraction. Whatever you multiply the numerator by, you must multiply the denominator by too. Check your answer by putting a number such as $x = 1$ into both the original and the simplified form.