Order of Operations Calculator

What Is the Order of Operations?

The order of operations is a set of mathematical rules that determines the correct sequence for evaluating an expression containing multiple operations. These rules make sure that the same expression produces the same result regardless of who solves it.

For example, consider the expression $3 + 4 \times 2$. If you add first, you get $14$, but if you multiply first, you get $11$. The order of operations tells us to perform multiplication before addition, so the correct answer is:

PEMDAS and BODMAS

PEMDAS is a common way to remember the order of operations. Different countries and textbooks may use names such as BODMAS or GEMDAS, but the underlying calculation rules are essentially the same.

• P — Parentheses: Evaluate expressions inside parentheses first.
• E — Exponents: Evaluate powers and other exponent operations next.
• M/D — Multiplication and Division: Perform these operations from left to right.
• A/S — Addition and Subtraction: Perform these operations from left to right.

In BODMAS, the letters commonly represent Brackets, Orders, Division, Multiplication, Addition, and Subtraction. The important point is that multiplication does not automatically come before division, and addition does not automatically come before subtraction. Operations at the same level are evaluated from left to right.

How to Solve an Expression Using the Order of Operations

Follow these steps when solving a mathematical expression:

1. 1. Solve expressions inside parentheses or brackets.
2. 2. Evaluate exponents and powers.
3. 3. Perform multiplication and division from left to right.
4. 4. Perform addition and subtraction from left to right.
5. 5. Continue until only the final value remains.

Worked Example: 12 + 4 × (8 - 3) ÷ 2

Let us evaluate the expression $12 + 4 \times (8 - 3) \div 2$ step by step using the order of operations.

Step 1: Solve the parentheses. First calculate $8 - 3$:

The expression becomes:

Step 2: Perform multiplication and division from left to right. The multiplication comes first because it appears before the division:

Now the expression is:

Next, divide:

Finally, perform the addition:

Therefore, the correct result is 22.

Example With Exponents

Exponents are evaluated before multiplication, division, addition, or subtraction. Consider:

First evaluate the exponent:

Then multiply:

Finally:

Order of Operations With Fractions

The same order of operations applies when an expression contains fractions. Parentheses, exponents, multiplication, division, addition, and subtraction must still be handled in the correct sequence.

For example:

First perform the multiplication:

Then add:

So the value of the expression is 2.

Order of Operations With Negative Numbers

Negative numbers can make expressions more difficult to read, especially when parentheses are involved. Always pay attention to whether the negative sign belongs to a number or is part of a subtraction operation.

For example:

Under the standard convention, the exponent is evaluated before the leading negative sign, so $-3^2 = -(3^2) = -9$. Therefore:

However, if the expression is $(-3)^2$, the parentheses make the negative number part of the base:

This is why parentheses are important when working with negative numbers and exponents.

Multiplication and Division: Work From Left to Right

A common misunderstanding is that multiplication must always be performed before division. In fact, multiplication and division have equal precedence. They are evaluated from left to right.

Consider:

Start with the division because it appears first:

Then multiply:

Therefore, $24 \div 6 \times 2 = 8$, not 2.

Addition and Subtraction Also Use Left-to-Right Order

Addition and subtraction also have equal precedence. When they occur together without parentheses, evaluate them from left to right.

For example:

First subtract:

Then add:

Common Mistakes to Avoid

• Adding before multiplying: Multiplication and division must be completed before addition and subtraction.
• Always multiplying before dividing: Multiplication and division have equal precedence and are evaluated from left to right.
• Always adding before subtracting: Addition and subtraction also have equal precedence and are evaluated from left to right.
• Ignoring parentheses: Expressions inside parentheses must be evaluated first.
• Misreading negative exponents or signs: Pay careful attention to parentheses around negative numbers.
• Skipping intermediate steps: Writing each step makes complex expressions easier to check and reduces calculation errors.

Why Use an Order of Operations Calculator?

An Order of Operations Calculator can evaluate complex mathematical expressions while following the correct order of operations. It is useful for checking homework, verifying calculations, simplifying expressions, and understanding how different operations are applied.

A calculator with step-by-step working is especially helpful when an expression contains parentheses, fractions, exponents, negative numbers, multiplication, division, addition, and subtraction. Instead of giving only the final answer, the intermediate steps can show how the expression was evaluated.