What does it mean to evaluate an algebraic expression?

To evaluate an algebraic expression means to replace each variable (letter) with its given numerical value and then simplify the result using the order of operations. The answer is a single number.

Why should I use parentheses when substituting values?

You should always put substituted values in parentheses, especially negative values. This avoids sign errors. For example, if x = -3, write (-3) instead of -3. Parentheses also clarify operations like powers: (-4)^2 = 16, but -4^2 = -16.

How do I evaluate an expression with negative numbers and powers?

Substitute the negative value inside parentheses, then apply the power. For example, to evaluate x^2 when x = -3: x^2 = (-3)^2 = 9. The parentheses ensure the negative sign is included in the squaring operation.

Evaluating Expressions

Learn how to evaluate algebraic expressions by substituting values for the variables and then simplifying. This page covers the key rules — including how to handle negatives and powers correctly — and walks through four worked examples.

What does "Evaluate" Mean?

To evaluate an algebraic expression, you replace each variable (letter) with its given numerical value, then simplify the result using the order of operations (PEMDAS). The final answer is a single number.

The key skill is careful substitution: drop the value in cleanly, watch the signs, and apply the rules of arithmetic in the right order.

Key rules when substituting

Drop-in with parentheses: if $x = - 3$ , write $(- 3)$ , not just $- 3$ . So $2 x$ becomes $2 (- 3) = - 6$ .
Powers of negatives: $(- 4)^{2} = (- 4) \cdot (- 4) = 16$ , but $- 4^{2} = - (4 \cdot 4) = - 16$ . The parentheses matter.
Multiplying signs: negative × negative = positive; negative × positive = negative.
Implied multiplication: $a b c$ means $a \cdot b \cdot c$ ; $3 e f$ means $3 \cdot e \cdot f$ .
Fractions: evaluate the numerator and denominator separately, then divide.
Formulas: identify which letter each given value replaces, then substitute and simplify.

Order of operations (PEMDAS)

After substituting, simplify using the standard order of operations, often remembered as PEMDAS (Please Excuse My Dear Aunt Sally):

Parentheses — work out anything in parentheses first.
Exponents — powers and roots next.
Multiplication and Division — left to right.
Addition and Subtraction — left to right.

Get any of these out of order and the answer will be wrong, even if your substitution was correct.

⚠️ Watch out: sign and parenthesis errors

The most common mistakes when evaluating are:

Forgetting parentheses around a negative. Substituting $x = - 2$ into $x^{2}$ without parentheses gives $- 2^{2} = - 4$ (wrong). With parentheses: $(- 2)^{2} = 4$ (correct).
Wrong order of operations. $3 + 4 \cdot 2 = 11$ , not $14$ . Multiply before adding.
Sign mistakes with subtraction. $5 - (- 3) = 5 + 3 = 8$ , not $2$ . Two negatives become a plus.

Worked examples

Example 1 — Multiplying with a Negative

Evaluate $3 p q$ when $p = 4$ and $q = - 5$ .

Solution

Substitute each value with parentheses, then multiply left to right:

\begin{aligned} 3 p q & = 3 \cdot 4 \cdot (- 5) \\ = 12 \cdot (- 5) \\ = - 60 \end{aligned}

Example 2 — Powers and Subtraction

Evaluate $m^{2} - 4 n$ when $m = - 3$ and $n = - 2$ .

Solution

Use parentheses around the negative values, especially when squaring:

\begin{aligned} m^{2} - 4 n & = (- 3)^{2} - 4 (- 2) \\ = 9 + 8 \\ = 17 \end{aligned}

Example 3 — Fractions

Evaluate $\frac{3 x + y}{z}$ when $x = 4$ , $y = - 2$ and $z = 5$ .

Solution

Evaluate the numerator first, then divide:

\begin{aligned} \frac{3 x + y}{z} & = \frac{3 (4) + (- 2)}{5} \\ = \frac{12 - 2}{5} \\ = \frac{10}{5} = 2 \end{aligned}

Example 4 — Using a Formula

The area of a rectangle is given by $A = l w$ . Find $A$ when $l = 12$ cm and $w = 7$ cm.

Solution

Substitute the given lengths into the formula:

\begin{aligned} A & = l \cdot w \\ = 12 \cdot 7 \\ = 84 {cm}^{2} \end{aligned}

Key rules to remember

Substitute every variable in parentheses, especially if the value is negative. This single habit prevents most sign errors.
Apply the order of operations (PEMDAS): parentheses, exponents, multiplication and division, then addition and subtraction.
Watch the powers of negatives: $(- 4)^{2} = 16$ , but $- 4^{2} = - 16$ . The parentheses change the meaning.
For fractions, evaluate the numerator and denominator separately, then divide.
For formulas, match each given value to the right letter before substituting.
Double-check signs at the end. Many wrong answers come down to a missed negative.
Show your working line by line. Substitution, then simplify step by step — don't try to do it all in your head.

Evaluating Expressions

Theory

What does "Evaluate" Mean?

Example 1 — Multiplying with a Negative

Example 2 — Powers and Subtraction

Example 3 — Fractions

Example 4 — Using a Formula

Video Lessons

Practice Questions

Evaluating Expressions

📖 Theory

What does "Evaluate" Mean?

Example 1 — Multiplying with a Negative

Example 2 — Powers and Subtraction

Example 3 — Fractions

Example 4 — Using a Formula

🎬 Video Lessons

✏️ Practice Questions

Theory

Video Lessons

Practice Questions