How to simplify expressions and complex fractions

To simplify an expression is to write it in a shorter, equivalent form — its value doesn’t change for any $x$. The tools are always the same: expand brackets, collect like terms, apply the exponent rules, and cancel common factors. This lesson works through each, then shows how to simplify complex fractions, where a fraction sits inside another fraction.

Simplify the following expression: like terms

Simplifying with exponent rules

• $x^a \cdot x^b = x^{a + b}$ — add exponents when multiplying.
• $\frac{x^a}{x^b} = x^{a - b}$ — subtract when dividing.
• $(x^a)^b = x^{ab}$ — multiply for a power of a power.
• $x^{-n} = \frac{1}{x^n}$ and $x^0 = 1$ (for $x \neq 0$).

For example, $\frac{6x^5 \cdot 2x^2}{4x^3} = \frac{12x^7}{4x^3} = 3x^4$.

How to simplify complex fractions

A complex fraction has a fraction in its numerator, its denominator, or both. The quickest method is to multiply the top and the bottom by the lowest common denominator of all the small fractions — that clears them in one step.

Common mistakes

• Combining unlike terms: $3x + 2x^2$ is already simplified.
• Forgetting to distribute a minus sign: $-x(x + 4) = -x^2 - 4x$.
• Cancelling terms rather than factors: $\frac{x + 2}{x}$ is not $2$.
• Multiplying only the top of a complex fraction by the common denominator.