Simplify an Expression
Simplifying an expression means rewriting it in its most compact form by combining like terms, applying properties, and removing unnecessary operations.
Definition
Simplifying an expression means combining everything you can, so it becomes as short as possible without changing its value: applying the distributive property to remove parentheses, then collecting and combining all like terms so no variable term appears more than once. Simplifying is not solving; after simplifying, the expression still contains variables, and solving would require an equation with an equals sign and a value for the variable. More formally, simplification is the process of applying algebraic identities and rewrite rules to produce a canonical or reduced form of an expression; in computer algebra systems this relies on normal forms such as expanded polynomial form or rational canonical form, and what counts as "simplest" depends on context, since factored form can be simpler than expanded form for some purposes.
Example
Simplify $3x + 5 + 2x - 1$: combine the $x$ terms ($3x + 2x = 5x$) and the constants ($5 - 1 = 4$) to get $5x + 4$. Simplify $2(3x - 4) + 5x$: distribute to get $6x - 8 + 5x$, then combine like terms to get $11x - 8$. Simplifying $(x^2-9)/(x-3)$ involves factoring the numerator, $(x+3)(x-3)/(x-3) = x + 3$, with the restriction $x \neq 3$.
Key Insight
A simplified expression is easier to evaluate and work with, like tidying up a messy room so it is easier to navigate. No single universal definition of "simplest" exists in algebra though: computer algebra systems use specific normal forms and heuristics, which is why different systems can produce different-looking but equivalent results.