Expressions, Substitution, and Exponent Rules

Algebraic Expressions, Substitution, and Order of Operations

An algebraic expression can contain numbers, variables, and operation symbols; a single variable or constant can also be an expression. A key feature of an expression is that it has no equal sign. In contrast, an equation uses an equal sign to set two expressions equal to one another. When working with expressions, you do not solve for an unknown value; instead, you simplify the expression or evaluate it.

Evaluating an expression means replacing variables with given numbers and calculating the final numerical value. Placing parentheses around every substituted number is a powerful habit that prevents common calculation errors, particularly when working with negative values. For example, if an expression contains a squared variable and you substitute a negative value, writing the number inside parentheses ensures that the entire negative quantity is squared properly.

Evaluating any expression requires following the standard order of operations. First, calculate all grouped expressions. Grouping symbols include standard parentheses, but they also include absolute value bars and fraction bars. Absolute value bars require you to evaluate the interior expression before determining the distance from zero. Similarly, a fraction bar acts as a grouping symbol, meaning the entire numerator and the entire denominator must each be evaluated before dividing. After grouped expressions are resolved, calculate exponents, then carry out multiplication and division from left to right, and finally perform addition and subtraction from left to right.

A critical distinction arises when negative numbers are squared. When a negative number is enclosed in parentheses and squared, such as (-4)², the negative value is multiplied by itself, producing positive 16. Without parentheses, an expression like -4² evaluates to negative 16 because the exponent applies only to the base 4, leaving the negative sign out front.