Dakota Prep

All math expressions and equations are solved universally with a set of rules called PEMDAS. Notice that the MD and DM are interchangeable, as the AS and SA. Let’s break this down to understand why.

Proper use of PEMDAS will result in the correct solution - hence it is the universally agreed upon Order of Operations.

Here is a fun video that can help you remember the order of operations:

‍

PEMDAS means:

P → Parentheses (work inside brackets or parentheses first)
E → Exponents (squares, cubes, etc.)
M/D → Division and Multiplication, interchangeable within operations but solve left to right if its a mix of multiplication and division (for example, 2 3 is the same as 32)
A/S → Addition and Subtraction, in order from left to right but also interchangeable (for example, 1 - 3 is the same as -3 +1)
‍

Note that once you reach an equation that is only multiplication & division or only addition and subtraction, you solve left to right.

Here is an example of why PEMDAS is important, and the way we write equations is important. There is an ongoing online debate regarding the answer to this question:

‍

8÷2(2+2)=?

‍

Here is an article discussing how it “broke the internet”.

The main issue here is once we solve inside the parentheses, we are left with 8÷2(4) and it is unclear whether to multiply the 2 and 4 first, or divide 8 by 2 first. This would give two answers. The first way would result in 1 and the second would leave 16. The correct answer is 16 because, even though it looks tempting to multiply the 2 and 4 first because they are next to each other, it must actually be solved left to right as they are all division and multiplication.

So, it should say 824 which, solved from left to right, would equal 1.

Here are some other examples of how to use PEMDAS.

‍

Example 1

‍

Simplify: 3x² + 4(x + 2)² when x = 3

‍

Step 1: Plug in the value of x
Anytime you’re told “when x = 3,” start by replacing x with 3.
So: 3(3²) + 4(3 + 2)²

‍

Step 2: Parentheses
Inside the parentheses: (3 + 2) = 5.
Now it’s 3(3²) + 4(5²)

‍

Step 3: Exponents
Do the squares: 3² = 9 and 5² = 25.
So now: 3(9) + 4(25)

‍

Step 4: Multiplication
Work those out: 3 × 9 = 27, 4 × 25 = 100.
So: 27 + 100

‍

Step 5: Addition
27 + 100 = 127

‍

Final Answer: 127

Teacher tip: The biggest mistake here is doing multiplication before the exponent. Always finish the exponent step before multiplying.

‍

Example 2

‍

Simplify: 6(2³) – 4(2 + 1)²

‍

Step 1: Parentheses
Look inside the parentheses first.
(2³) stays as it is for now.
(2 + 1) = 3.
So we get: 6(2³) – 4(3)²

‍

Step 2: Exponents
2³ = 8.
3² = 9.
So now: 6(8) – 4(9)

‍

Step 3: Multiplication
6 × 8 = 48.
4 × 9 = 36.
So: 48 – 36

‍

Step 4: Subtraction
48 – 36 = 12.

‍

Final Answer: 12

Teacher tip: Notice how the parentheses changed the order. If you forgot them and tried to just run left to right, you’d get the wrong result.

‍

Example 3

‍

Simplify: –12 + (18 – 6) ÷ 3

‍

Step 1: Parentheses
Inside (18 – 6) = 12
So: –12 + 12 ÷ 3

‍

Step 2: Division
12 ÷ 3 = 4
So: –12 + 4

‍

Step 3: Addition
–12 + 4 = –8.

‍

Final Answer: –8

Teacher tip: Be mindful of negatives. Many students accidentally make it +12 instead of –12.

‍

Common Mistakes to Watch Out For

‍

Doing multiplication before exponents.
Example: In 3(2²), you must square 2 first (2² = 4), then multiply by 3. Don’t do 3×2 first.
Forgetting the left-to-right rule.
In 20 ÷ 5 × 2, you go left to right: (20 ÷ 5 = 4), then (4 × 2 = 8). If you did multiply first, you’d get the wrong answer.
Dropping negative signs.
Remember, –12 is not the same as +12. Carry the negative through each step.
Skipping steps.
Write out each stage. If you try to do everything in your head, mistakes pile up fast.

Quick “PEMDAS” Practice (try these)

‍

5 + 3 × 2
(7 – 3)² + 4
18 ÷ (6 – 3) × 2
–4² + 10
2(3 + 5)² – 6
‍

Practice Answers:

5 + 6 = 11
4² + 4 = 20‍
18 3 2 = 12
–6 (because –(4²) = –16, then add 10)
2 8² - 6 = 122

‍