By Chris Deziel, Updated Aug 30, 2022

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Take a look at the following equation:

\(x = 7 + 2 \times (11 - 5) \div 3\)

If you simply read the expression from left to right you’ll arrive at 18, which is incorrect. The correct result—11—comes from applying the standard order of operations, often remembered by the acronym PEMDAS:

• P – Parentheses
• E – Exponents
• M – Multiplication
• D – Division
• A – Addition
• S – Subtraction

For those who find PEMDAS hard to recall, mnemonic phrases such as "Please Excuse My Dear Aunt Sally" or the alternative BEDMAS (“Big Elephants Destroy Mice and Snails”) can be useful. The key is that multiplication and division are performed in the order they appear, as are addition and subtraction.

Applying the Order of Operations

When confronted with a long string of operations, the mathematician’s rulebook is straightforward:

1. Resolve all expressions inside parentheses or brackets, working from the innermost out.
2. Solve any exponents.
3. Carry out multiplication and division from left to right.
4. Finish with addition and subtraction from left to right.

Sample Calculation

Let’s solve the earlier equation step by step.

1. Handle the Parentheses

\(11 - 5 = 6\)

The expression becomes:

\(x = 7 + 2 \times 6 \div 3\)

2. Perform Multiplication and Division

First, multiplication:

\(2 \times 6 = 12\)

Then division:

\(12 \div 3 = 4\)

The equation now reads:

\(x = 7 + 4\)

3. Finish with Addition

\(7 + 4 = 11\)

Thus, x = 11.

Additional Examples

1. 15 – [5 + (7 - 4)]

1. Inner parentheses: 7 - 4 = 3
2. Outer brackets: 5 + 3 = 8
3. Subtraction: 15 - 8 = 7

2. (5 - 3)^2 + (10 ÷ [7 - 2])^2 × 4

1. Inner parentheses: 5 - 3 = 2 and 7 - 2 = 5
2. Exponents: 2^2 = 4 and (10 ÷ 5)^2 = 2^2 = 4
3. Multiplication: 4 × 4 = 16
4. Addition: 4 + 16 = 20

Remember: whether you encounter parentheses, brackets, or braces, always tackle the innermost expressions first, then follow PEMDAS to reach the final answer confidently.