GEMDAS, and the three places it is usually misread.
Evaluate
12 − 3 × 2² + (8 − 5)
Brackets and powers first: (8 − 5) = 3 and 2² = 4. Then the multiplication: 3 × 4 = 12. Finally left to right: 12 − 12 + 3 = 3. D works straight left to right and subtracts before multiplying.
| Rank | Do | Note |
|---|---|---|
| 1 | Grouping — brackets | innermost first |
| 2 | Exponents | powers and roots |
| 3 | Multiply and Divide | same rank, left to right |
| 4 | Add and Subtract | same rank, left to right |
Multiplication does not outrank division, and addition does not outrank subtraction. Reading GEMDAS as six separate stages is where most errors start.
12 − 3 × 2² + (8 − 5)
the expression
12 − 3 × 2² + 3
brackets first
12 − 3 × 4 + 3
the exponent
12 − 12 + 3
multiply
0 + 3 = 3
left to right
Doing 12 − 12 + 3 right to left gives 12 − 15 = −3. Direction matters.
−(3 × 3) = −9
the power binds tighter
(−3)(−3) = 9
the bracket takes the sign
Without brackets the exponent applies to the 3 alone, and the minus is applied afterwards.
Evaluating x² at x = −3 means (−3)² = 9, because x is −3. Typing −3² into a calculator returns −9, which is a different question.
| Expression | Correct | Common slip |
|---|---|---|
| 24 ÷ 4 × 2 | 12 | 3 |
| 20 − 8 + 5 | 17 | 7 |
| 36 ÷ 6 ÷ 3 | 2 | 18 |
| 100 ÷ 5 × 2 ÷ 4 | 10 | 2.5 |
| 15 − (−6) + 2 | 23 — subtracting −6 adds 6 | 11 |
The first three slips group the later operation first: 24 ÷ (4 × 2), 20 − (8 + 5), 36 ÷ (6 ÷ 3). The last one pulls the × 2 under the division: 100 ÷ (5 × 2 × 4) = 100 ÷ 40 = 2.5. Read like a sentence — left to right.
A fraction bar, a root sign and an exponent all group silently.
3 + 92 + 4
the bar groups both lines
= 12 ÷ 6 = 2
not 3 + 4.5 + 4
√9 + 16 = √25 = 5
the root groups too
Typing that fraction into a calculator needs real brackets: (3 + 9) ÷ (2 + 4). Without them it works out 3 + 4.5 + 4 and returns 11.5.
Brackets inside brackets: 2{18 − 3[4 + (1 − 3)³]}.
(1 − 3)³ = (−2)³ = −8
innermost first, then its power
[4 + (−8)] = −4
next layer out
{18 − 3(−4)} = 18 + 12 = 30
multiply before subtracting
2 × 30 = 60
outermost last
What is different: three layers. Work from the inside out and rewrite the line after each layer.
The same order with fractions: 12 + 34 × 23.
34 × 23 = 612 = 12
the multiplication first
12 + 12 = 1
then the addition
What is different: fractions tempt you to add first because adding them feels like the “real” work. The ranks do not change.
Adding first gives (12 + 34) × 23 = 54 × 23 = 56. That puts brackets where there are none.
A power on a product: 2 × 3² against (2 × 3)².
2 × 3² = 2 × 9 = 18
the power takes the 3 only
(2 × 3)² = 6² = 36
the bracket squares both factors
What is different: the same idea as −3² against (−3)², with a number in place of the minus sign.
Finish everything inside the bars, then take the size (drop the sign), then carry on outside. Evaluate 10 − 2|1 − 3²| + |−6| ÷ 3.
inside: 1 − 3² = 1 − 9 = −8
the power before the minus
|−8| = 8, |−6| = 6
now drop the signs
10 − 2 × 8 + 6 ÷ 3
bars gone, ordinary order
10 − 16 + 2 = −4
multiply and divide, then left to right
The bars do not make the whole answer positive: the − 2 in front still subtracts 16. And they do not split: with x = −2, |x − 5| = |−7| = 7, but |x| − |5| = 2 − 5 = −3. Work the inside to one number before you drop any sign.
These items pack brackets, a power of a negative, a fraction bar and absolute-value bars into one line, and each choice is one slip away from the answer. Rewrite the line once after every stage, and check the two classics: (−3)² against −3², and ÷ then × from left to right.
Evaluate
18 ÷ 3 × 2 + 5 − 4²
Division and multiplication rank equally, so go left to right: 18 ÷ 3 = 6, then 6 × 2 = 12. With 4² = 16, that leaves 12 + 5 − 16 = 1. A does 3 × 2 first — reading "MD" as multiply-before-divide is the trap.
