Algebra · Evaluate Expressions
Order of operations is a shared convention for reading an expression. Its most important detail is easy to miss: multiplication and division have equal priority, as do addition and subtraction.
Simplify grouping symbols first, then powers and roots. Next, work multiplication and division from left to right. Finally, work addition and subtraction from left to right.
PEMDAS and BODMAS are memory aids for this same priority structure; they do not mean multiplication always beats division. For example, 18 ÷ 3 × 2 = 12, not 3. OpenStax states the left-to-right rule explicitly.
What Is The Correct Order Of Operations?
Use four levels, not six unrelated steps. First evaluate what is inside parentheses, brackets, or another grouping symbol, starting with the innermost group. Then evaluate exponents or roots.
Multiplication and division share the next level; addition and subtraction share the final level. Within either shared level, read the expression from left to right. OpenStax’s procedure follows these four levels.
In PEMDAS, the letters are Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. In BODMAS, the corresponding terms are Brackets, Orders, Division, Multiplication, Addition, Subtraction. The different letter order does not change the equal-priority pairs.
Treat M and D as one level, and A and S as one level.
How Do You Evaluate An Expression Step By Step?
Consider 4 + 3 × (8 − 5)2. Simplify inside the parentheses first: 8 − 5 = 3. Then square 3 to get 9.
Multiply 3 × 9 = 27. Add 4 to get 31. The subtraction inside the parentheses happens before multiplication because the parentheses explicitly group it.
| Stage | Expression | Reason |
|---|---|---|
| Start | 4 + 3 × (8 − 5)2 | Read the grouped part |
| Group | 4 + 3 × 32 | 8 − 5 = 3 |
| Power | 4 + 3 × 9 | 32 = 9 |
| Multiply | 4 + 27 | 3 × 9 = 27 |
| Add | 31 | 4 + 27 = 31 |
Write one changed stage per line. That makes it easier to locate a mistake than jumping directly from the original expression to a result. The scientific calculator is useful for checking a final value, while the written stages explain why it is correct.
Why Does Left To Right Matter?
Multiplication does not automatically come before division. For 18 ÷ 3 × 2, divide 18 by 3 to get 6, then multiply 6 by 2 to get 12. Doing 3 × 2 first would silently insert parentheses that are not present.
The same principle applies to subtraction and addition: 10 − 4 + 2 = 8, because 10 − 4 = 6, then 6 + 2 = 8.
How Do Grouping Symbols Change The Result?
Parentheses tell you to calculate a particular part first. Compare 4 + 3 × 5 = 19 with (4 + 3) × 5 = 35. The digits and operations are the same, but the parentheses change which operation occurs first.
A fraction bar can also group its entire numerator and denominator; write those parts clearly if you are typing a fraction into a calculator.
Example: Make The Grouping Explicit
If you mean “six plus two, then divide by four,” write (6 + 2) ÷ 4 = 2. Writing 6 + 2 ÷ 4 means 6 + 0.5 = 6.5. When an expression could be read in more than one way in plain text, parentheses make the intended calculation unmistakable.
Nested grouping works from the inside outward. In 2 × [3 + (8 − 5)], calculate 8 − 5 = 3, then 3 + 3 = 6, then 2 × 6 = 12. OpenStax recommends beginning with the innermost group.
Why Does A Fraction Bar Act Like Grouping?
A fraction bar groups its full numerator and full denominator. For (6 + 2)/(5 − 1), evaluate the grouped amounts to get 8/4 = 2. A typed slash needs parentheses to preserve that same structure.
Writing 6 + 2/5 − 1 instead gives 5.4. That is a different expression, not another valid answer to the grouped fraction. The layout of notation supplies mathematical information.
Convert a stacked fraction into an explicit parenthesized input before checking it with a tool. Do not assume the tool can infer the grouping from an incomplete text string.
How Do Parentheses Affect A Negative Base?
In (−3)², the whole negative number is the base. Multiplying −3 by −3 gives 9. In −3², the power applies to 3 first, and the external minus sign then gives −9.
For (−3)³, the result is −27 because three negative factors produce a negative product. The expression −3³ happens to give −27 as well, but that agreement does not make the grouping irrelevant. Even powers expose the difference.
Write parentheses around a negative value when substituting it into a power. This preserves the variable’s role as the base and prevents a correct algebra result from appearing inconsistent.
Can Algebraic Rewriting Change The Evaluation Order?
A valid identity can rewrite an expression into an equivalent form. For example, 2(3 + 4) equals 2 × 3 + 2 × 4. Both evaluate to 14.
The second form no longer has the original parentheses because distribution has preserved their effect explicitly. This does not contradict the grouping rule. It replaces the expression with a proven equivalent expression.
Do not make the same move with a false identity. The expression (3 + 4)² is 49, while 3² + 4² is 25. A square of a sum includes a middle product term.
How Should You Handle Ambiguous Plain-Text Expressions?
Rewrite the intended grouping instead of arguing from a memory aid. Expressions with a slash followed by adjacent multiplication can be read differently under different notation conventions. Parentheses or a full fraction bar make the meaning clear.
If you intend the whole product to be the denominator, write a/(bc). If you intend a divided by b, then multiplied by c, write (a/b)c. These are distinct expressions and generally give different values.
Ask for clarification when the source does not establish which one was intended. A confident calculator output cannot resolve an ambiguity in the question itself.
Does Left To Right Mean Every Operation Is Done Immediately?
No. Left-to-right evaluation applies within the stated equal-priority level. In 2 + 3 × 4, multiplication still comes before addition, even though the plus sign appears first.
In 20 − 8 ÷ 2 + 1, evaluate 8 ÷ 2 = 4 first. Then perform 20 − 4 + 1 from left to right, giving 17. Reading every symbol immediately from the start would produce a different, incorrect value.
FAQs About Order Of Operations
Do PEMDAS And BODMAS Define Different Arithmetic?
No. They label the same conventional priority groups using different words. The multiplication/division and addition/subtraction pairs still share priority.
What Should I Do If A Calculator Result Differs?
Check the complete input, grouping, negative bases, and rounding before assuming an arithmetic error. Confirm that both calculations evaluate the same expression, not merely the same visible digits.
Can You Check Your Understanding?
Try these without a calculator first. Say which priority rule you used at every stage, then compare with the answers.
Show the Worked Answers
6 + 4 × (7 − 2)2 ÷ 5: the group is 5, its square is 25, then 4 × 25 ÷ 5 = 20 from left to right, and 6 + 20 = 26.
24 ÷ 6 × 2 − 3: 24 ÷ 6 = 4, 4 × 2 = 8, and 8 − 3 = 5.
If an answer differs, first check the equal-priority operations. Working them in the wrong order is more common than forgetting that parentheses come first. For further practice, visit SolverCalc Math Practice.
Check An Expression
Enter the expression with explicit parentheses, compare the result, and then review your written steps.
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