How Do You Do Pemdas? | Order of Operations Rules

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You follow the specific sequence: Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right).

Math problems can turn into chaotic guessing games without a standard set of rules. You might look at a string of numbers and symbols, unsure where to begin. If you subtract before you multiply, or add before you solve the exponents, you end up with a completely wrong answer. This confusion is why the order of operations exists.

Students and adults alike often stumble on viral math problems because they forget the hierarchy of calculation. The acronym PEMDAS serves as the roadmap. It dictates exactly which part of an equation gets priority so that everyone, everywhere, calculates the same result.

This guide explains the rules, breaks down the tricky “left-to-right” exceptions, and walks you through complex examples.

What Does PEMDAS Stand For?

PEMDAS is an acronym used primarily in the United States to help people remember the order of operations. Each letter represents a mathematical operation. You must address these operations in a specific rank order to solve an equation correctly.

Here is the breakdown of the hierarchy:

  • P — Parentheses: Solve everything inside grouping symbols first. This includes brackets [] and braces {}.
  • E — Exponents: Calculate powers and square roots next.
  • M — Multiplication: Multiply numbers. (Ranked equally with Division).
  • D — Division: Divide numbers. (Ranked equally with Multiplication).
  • A — Addition: Add numbers. (Ranked equally with Subtraction).
  • S — Subtraction: Subtract numbers. (Ranked equally with Addition).

Many people mistake this list for a strict six-step process where multiplication always happens before division. That is incorrect. The relationship between multiplication/division and addition/subtraction is fluid based on their position in the equation.

How Do You Do Pemdas? – The Step-by-Step Method

Applying the rules requires a systematic approach. You scan the problem from left to right, but you only engage with the specific operation that has the highest priority at that moment.

Quick check: If you see a math problem with multiple operations, stop and identify the groups. Do not just start adding the first two numbers you see.

Follow this exact process to solve any linear math equation:

  1. Scan for grouping symbols — Look for parentheses or brackets. Solve the math problem inside them until you have a single number remaining.
  2. Resolve the exponents — Find any small floating numbers (powers) or radical signs (roots) and calculate their value.
  3. Handle multiplication and division — Scan the problem from left to right. Whichever operator comes first, do that one. If division is on the left of multiplication, divide first.
  4. Finish with addition and subtraction — Scan again from left to right. Treat these as equals. If subtraction appears before addition, subtract first.

Learning how do you do Pemdas correctly requires practice with this specific scanning method. You cannot skip steps or jump ahead just because the addition looks easier than the division.

The “Left-To-Right” Rule Explained

The most common source of errors comes from the “MD” and “AS” parts of the acronym. M comes before D in the word PEMDAS, so people assume multiplication is always superior. This is a myth. Multiplication and Division are “inverse operations,” meaning they hold the same weight/rank.

Multiplication and Division Priority

Think of these two as a married couple with equal standing. When you reach the step to multiply or divide, you simply do whichever one appears first as you read the equation like a book.

Consider the problem: 20 ÷ 4 × 2

  • Incorrect method — If you multiply 4 × 2 first, you get 8. Then 20 ÷ 8 = 2.5. This is wrong because you prioritized multiplication just because of the acronym letter order.
  • Correct method — Scan left to right. The division (÷) happens first. 20 ÷ 4 = 5. Now take that 5 and multiply by 2. The answer is 10.

Addition and Subtraction Priority

The same logic applies to the final two letters. Addition does not inherently beat subtraction. They are also inverse operations.

Consider the problem: 10 – 4 + 3

  • Incorrect method — If you add 4 + 3 first, you get 7. Then 10 – 7 = 3. This changes the value of the numbers and yields a false result.
  • Correct method — Scan left to right. Subtraction appears first. 10 – 4 = 6. Now take that 6 and add 3. The answer is 9.

Handling Parentheses And Grouping Symbols

The “P” in PEMDAS covers more than just curved parentheses `()`. It dictates that you must simplify grouped terms before doing anything else with the outside numbers. This includes square brackets `[]`, curly braces `{}`, and even fraction bars.

When you encounter a fraction bar, the entire numerator (top) is a group, and the entire denominator (bottom) is a group. You must solve the top and bottom completely before you divide the top by the bottom.

Working With Nested Parentheses

Sometimes you will see parentheses inside of brackets. This looks intimidating, but the rule is simple: work from the inside out.

Deeper fix: Find the innermost set of parentheses. Solve that equation first. Then move to the brackets surrounding that result. Continue expanding outward until all grouping symbols are removed.

Example: 4 × [15 – (2 + 3)]

  1. Identify the inner group — The innermost math is (2 + 3). Solve this to get 5.
  2. Rewrite the equation — Now you have 4 × [15 – 5].
  3. Solve the bracket — The operation inside the bracket is 15 – 5, which equals 10.
  4. Final multiplication — Now you have 4 × 10. The final answer is 40.

Common Mistakes With The Order Of Operations Rules

Even advanced students make simple calculation errors by rushing. Identifying these pitfalls helps you avoid them on tests or in practical applications.

The “Implicit Multiplication” Trap

A number sitting directly next to a parenthesis implies multiplication. For example, `2(3)` is the same as `2 × 3`. This often confuses people when combined with division.

Take the viral math problem: 6 ÷ 2(1 + 2)

Many people add 1 + 2 to get 3, leaving 6 ÷ 2(3). Then, they multiply 2 × 3 to get 6, and finally divide 6 ÷ 6 to get 1. This is incorrect. The multiplication implied by `2(3)` does not have higher priority than the division symbol before it.

Correct path:

  • Parentheses first — 1 + 2 = 3. Equation is now 6 ÷ 2(3).
  • Left to right — Division comes first. 6 ÷ 2 = 3.
  • Final multiply — 3 × 3 = 9.

Forgetting Negative Numbers

When you square a negative number, parentheses matter. `-3²` is different from `(-3)²`.

  • Without parentheses — `-3²` means “the negative of 3 squared.” You square 3 (getting 9) and apply the negative. Result: -9.
  • With parentheses — `(-3)²` means “negative 3 times negative 3.” Result: 9.

Real-World Examples Of PEMDAS

Let’s apply the method to a few more scenarios to solidify your understanding. Seeing the step-by-step breakdown helps visual learners grasp the sequence.

Example 1: Mixed Operations

Problem: 5 + 18 ÷ 3² – 2

  1. Check for Parentheses — None here. Move to Exponents.
  2. Calculate Exponents — 3² is 3 × 3, which is 9. The equation becomes 5 + 18 ÷ 9 – 2.
  3. Multiplication/Division — We see division (18 ÷ 9). Solve that to get 2. The equation is now 5 + 2 – 2.
  4. Addition/Subtraction — Left to right. 5 + 2 is 7. Then 7 – 2 is 5.
  5. Final Answer — 5.

Example 2: Complex Grouping

Problem: (4 + 3)² – 10 ÷ 2

  1. Parentheses — 4 + 3 = 7. Equation: 7² – 10 ÷ 2.
  2. Exponents — 7² = 49. Equation: 49 – 10 ÷ 2.
  3. Division — 10 ÷ 2 = 5. Equation: 49 – 5.
  4. Subtraction — 49 – 5 = 44.
  5. Final Answer — 44.

PEMDAS vs. BODMAS vs. GEMDAS

If you study math outside the United States, you might hear different acronyms. They all follow the same logical principles but use slightly different terminology for the operations.

BODMAS (UK, Australia, India)

This stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. The main difference is the word “Brackets” instead of Parentheses and “Orders” instead of Exponents. The rules regarding left-to-right processing for DM and AS remain identical.

GEMDAS or GEMS

Some teachers prefer GEMDAS (Grouping symbols) to explicitly include brackets and braces alongside parentheses. GEMS is a shorter version: Grouping, Exponents, Multiplication/Division, Subtraction/Addition. This version is helpful because it groups the inverse operations (M/D and S/A) together, reducing the confusion about which comes first.

Practical Tips For Math Success

Mastering the order of operations takes time. Use these tips to ensure your calculations remain accurate.

Write out every step. Do not try to hold the changing numbers in your head. Rewrite the entire line of the equation after solving one piece. This looks tedious, but it prevents 90 percent of errors.

Use the “Aunt Sally” phrase. The mnemonic “Please Excuse My Dear Aunt Sally” is the standard way to memorize the letters. If that feels outdated, invent your own. The sillier the phrase, the easier it is to recall during a test.

Check your work with a calculator. However, be careful. Basic calculators often perform operations immediately as you type them (chain calculation), ignoring PEMDAS. Scientific calculators typically follow the correct hierarchy. Knowing how do you do Pemdas manually allows you to spot when a calculator gives you a nonsensical result.

Key Takeaways: How Do You Do Pemdas?

➤ Prioritize grouping symbols first, working from the innermost set outward.

➤ Calculate exponents and roots immediately after clearing parentheses.

➤ Treat Multiplication and Division as equals; solve them left to right.

➤ Treat Addition and Subtraction as equals; solve them left to right.

➤ Rewrite each line after solving one step to maintain accuracy.

Frequently Asked Questions

Does multiplication always come before division?

No. In the PEMDAS acronym, M comes before D, but they share the same rank. You must solve them in the order they appear from left to right. If division appears first in the equation, you divide before you multiply.

What if there are no parentheses or exponents?

You simply skip the P and E steps and move directly to Multiplication and Division. Scan the problem for these operations. If those are also missing, jump straight to Addition and Subtraction. The hierarchy remains valid even if some steps are not present.

Why do calculators sometimes give different answers?

Basic “four-function” calculators often process numbers immediately as you press the keys. They do not “see” the whole equation. Scientific and graphing calculators are programmed with the order of operations built-in. Always know which type you are using to avoid calculation errors.

Do brackets and braces mean the same thing as parentheses?

Yes, mathematically they function as grouping symbols. You solve whatever is inside them first. Usually, parentheses `()` are the inner set, brackets `[]` surround those, and braces `{}` surround the brackets. You work from the inside shape to the outside shape.

Is GEMDAS better than PEMDAS?

GEMDAS is often considered clearer because “G” stands for “Grouping.” This reminds students that division bars and brackets are included, not just parentheses. However, both acronyms teach the exact same mathematical rules and result in the same answer.

Wrapping It Up – How Do You Do Pemdas?

Understanding the order of operations transforms math from a confusing jumble of numbers into a logical puzzle. By following the hierarchy—Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction—you ensure accuracy every time.

Remember the left-to-right rule for the inverse operations. This single detail is where most students get lost. Take your time, rewrite your work line by line, and double-check your steps. Once you master this foundation, algebra and complex equations become much easier to handle.