8.1 Simplifying & Evaluating Expressions

Variables, Constants, Coefficients, and Terms

Algebra extends arithmetic by using letters to stand for unknown or changing numbers. A variable is a symbol (usually a letter like x, y, or n) that represents an unknown value. A constant is a fixed number on its own, such as 7 or −3. A coefficient is the number multiplying a variable—for example, in 5x the coefficient is 5 and the variable is x. A term is any single number, variable, or product of numbers and variables; 8, y, 3x, and −2xy are all terms. An expression is a combination of terms connected by addition or subtraction, such as 3x + 5y − 7.

Understanding these pieces is the foundation for every other algebra skill on the NEX. When you simplify or evaluate, you are manipulating terms; when you solve an equation, you are isolating a variable term.

Combining Like Terms

Like terms are terms whose variable parts are identical—the same letter raised to the same power. 3x and 7x are like terms; 3x and 7x² are not, because the powers of x differ. 5y and −2y are like terms; 5y and 5x are not, because the variables differ. Constants are like terms with other constants.

To simplify an expression, combine like terms by adding or subtracting their coefficients and keeping the variable part unchanged.

Example 1. Simplify 4x + 9x − 2x.

All three terms are like terms (same variable x, same power 1). Add and subtract the coefficients: 4 + 9 − 2 = 11, so the result is 11x.

Example 2. Simplify 3x + 7y − 2x + 5y + 1.

Group the like terms: (3x − 2x) + (7y + 5y) + 1 = x + 12y + 1. The constant 1 has no like term, so it remains on its own.

Common trap: You cannot combine 3x and 7y just because they sit next to each other. 3x + 7y is already fully simplified. You also cannot combine 4x and 5x²—the exponents must match.

The Distributive Property

The distributive property states that a(b + c) = ab + ac. The factor outside the parentheses multiplies each term inside. The same rule works for subtraction: a(b − c) = ab − ac.

Example 3. Simplify 3(x + 4).

Multiply 3 by each term inside: 3·x = 3x and 3·4 = 12. The result is 3x + 12.

Example 4. Simplify −2(5x − 3).

Multiply −2 by each term inside: (−2)(5x) = −10x and (−2)(−3) = +6. The result is −10x + 6.

Common trap (sign when distributing a negative): When the outside factor is negative, the negative sign applies to every term inside the parentheses. A frequent mistake is writing −2(5x − 3) = −10x − 6, forgetting that (−2)(−3) is positive 6. Always carry the sign to each term and use the rules for multiplying signed numbers: same signs give a positive product, opposite signs give a negative product.

You can also distribute when there are three or more terms inside: 4(x² + 2x − 5) = 4x² + 8x − 20.

Combining Distribution with Like Terms

Many NEX problems ask you to simplify an expression that needs both steps—distribute first, then combine like terms.

Example 5. Simplify 2(x + 3) + 4x − 5.

Step 1: Distribute the 2. 2(x + 3) = 2x + 6. The expression becomes 2x + 6 + 4x − 5.

Step 2: Combine like terms. (2x + 4x) + (6 − 5) = 6x + 1.

Example 6. Simplify 3(2x − 1) − 2(x + 4).

Distribute both: 6x − 3 − 2x − 8. Notice the second distribution carries the negative: −2·x = −2x and −2·4 = −8. Now combine: (6x − 2x) + (−3 − 8) = 4x − 11.

Evaluating Expressions by Substitution

To evaluate an expression, substitute a given number for each variable and compute the result, following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division left to right, Addition/Subtraction left to right).

Example 7. Evaluate 3x + 7 when x = −2.

Substitute: 3(−2) + 7 = −6 + 7 = 1.

Example 8. Evaluate 2x² − 5x + 4 when x = 3.

Substitute: 2(3)² − 5(3) + 4. Follow order of operations: the exponent comes before the multiplication, so 3² = 9, giving 2·9 − 15 + 4 = 18 − 15 + 4 = 7. A common mistake is to compute 2·3 first and then square, which would give 36 instead of 18. Always apply the exponent to the variable value first.

Example 9. Evaluate 4(2x − 1) when x = 5.

You can substitute first: 4(2·5 − 1) = 4(10 − 1) = 4·9 = 36. Or simplify first using distribution: 4(2x − 1) = 8x − 4, then substitute: 8·5 − 4 = 40 − 4 = 36. Both paths give the same answer.

Basic Exponent Rules Inside Expressions

When the same variable appears with exponents, three rules let you simplify quickly. They work only when the base is the same variable.

1. Product rule: x^a · x^b = x^(a+b). When multiplying like bases, add the exponents. Example: x² · x³ = x^(2+3) = x⁵.
2. Power rule: (x^a)^b = x^(a·b). When raising a power to a power, multiply the exponents. Example: (x²)³ = x^(2·3) = x⁶.
3. Quotient rule: x^a / x^b = x^(a−b). When dividing like bases, subtract the exponents. Example: x⁵ / x² = x^(5−2) = x³.

Example 10. Simplify x⁴ · x².

Same base x, so add exponents: x^(4+2) = x⁶.

Example 11. Simplify (x³)² · x.

Apply the power rule first: (x³)² = x⁶. Then x⁶ · x = x⁶ · x¹ = x⁷.

Example 12. Simplify x⁶ / x².

Same base, so subtract exponents: x^(6−2) = x⁴.

Common trap: The rules above apply only to like bases. You cannot simplify x² · y³ with these rules because the bases differ. Also, exponents are not multiplied when bases are multiplied—x² · x³ = x⁵, not x⁶. Mixing up the product and power rules is a frequent error: x² · x³ adds exponents (giving x⁵), while (x²)³ multiplies them (giving x⁶). Know which operation is happening before choosing the rule.

Quick Self-Check

• 4x + 9 − 2x simplifies to 2x + 9 (combine 4x and −2x).
• −3(2x − 4) = −6x + 12, not −6x − 12 (sign on the constant).
• If x = 4, then 2x² − x = 2·16 − 4 = 28 (exponent before multiplication).
• x³ · x⁴ = x⁷ (add exponents), while (x³)⁴ = x¹² (multiply exponents).