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Order of operations
The same numbers can give four different answers depending on which operation you do first. The test lists all four. One fixed order, plus two details most people forget, gets you the right one every time.
One order, every time
- 8 + 2 × (7 − 3)²
- 8 + 2 × 4²
- 8 + 2 × 16
- 8 + 32
- 40
Everyone who writes math agrees on one order: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS). Work in that order and an expression has only one value.
The two details people forget
Multiplication and division are one step: do them left to right, whichever comes first. The same goes for addition and subtraction. PEMDAS does not mean “multiply before you divide.”
How to solve them
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1
The order: parentheses, exponents, × ÷, then + −
Do every step completely before you move to the next. Inside each step, go from left to right.
Parentheses ( ) [ ]1stwork inside them firstExponents and roots2ndx², √xMultiply and divide3rdleft to rightAdd and subtract4thleft to rightNext step: Left to right- 20 − 12 ÷ 4 × 2
- 20 − 3 × 2
- 20 − 6
- 14
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2
Ties go left to right
When multiplication and division sit side by side, the one on the left goes first. The same rule decides between addition and subtraction.
Next step: Nested brackets× and ÷18 ÷ 3 × 2
6 × 2 = 12, not 18 ÷ 6 = 3
+ and −10 − 4 + 3
6 + 3 = 9, not 10 − 7 = 3
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3
Brackets inside brackets: work from the inside out
Square brackets [ ] are just a second layer of parentheses. Clear the innermost pair first, then the next one out.
Next step: Exponents & signs- 3 × [10 − (2 + 4)]
- 3 × [10 − 6]
- 3 × 4
- 12
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4
Exponents come before multiplying, and the minus-sign trap
An exponent applies only to what it touches. In 5 × 2³ it touches the 2, so cube the 2 first. In −3² it touches the 3, not the minus sign: the answer is negative. Put the negative number in parentheses, (−3)², to square the whole thing.
Next step: Fraction barsExponent first5 × 2³
5 × 8 = 40, not 10³
No parentheses−3²
−(3 × 3) = −9
With parentheses(−3)²
(−3) × (−3) = 9
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5
A fraction bar groups the top and the bottom
A fraction bar works like two invisible pairs of parentheses: finish the whole top and the whole bottom, then divide.
(6 + 18) ÷ (2 × 4)
Next step: Write it out- (6 + 18) ÷ (2 × 4)
- 24 ÷ 8
- 3
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6
Write the whole line after every step
Most errors come from doing two steps in your head and losing a sign or a number. Rewrite the entire expression each time, changing one piece. It takes a few seconds and it is the habit that makes these questions free points.
ShortcutCircle the next operation
Before each step, find the operation PEMDAS says is next and circle it on your scratch paper. Do only that one, rewrite the line, and repeat.
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Ignoring left to right
24 ÷ 4 × 2:
Wrong24 ÷ 8 = 3
Right6 × 2 = 12
Adding before multiplying
5 + 3 × 4:
Wrong8 × 4 = 32
Right5 + 12 = 17
Squaring the minus sign
−4²:
Wrong16
Right−16
The exponent on the whole product
2 × 3²:
Wrong6² = 36
Right2 × 9 = 18
5 questions at test level
Pick an answer, then read why. No timer, no account.
Evaluate: 7 + 6 × 3 − 4
Show the answer
B. Multiply first: 6 × 3 = 18. Then left to right: 7 + 18 = 25 and 25 − 4 = 21. 35 adds 7 + 6 before multiplying. The value is 21.
Evaluate: 40 ÷ 5 × 2 + 3
Show the answer
C. Division and multiplication left to right: 40 ÷ 5 = 8, then 8 × 2 = 16. Add last: 16 + 3 = 19. 7 multiplies 5 × 2 first, and 16 forgets the + 3. The value is 19.
Evaluate: (9 − 3)² ÷ 4 + 1
Show the answer
B. Parentheses: 9 − 3 = 6. Exponent: 6² = 36. Divide: 36 ÷ 4 = 9. Add: 9 + 1 = 10. 19 squares the 9 and the 3 separately, and 7.2 adds 4 + 1 before dividing. The value is 10.
Evaluate: −2² + 3 × (4 − 6)
Show the answer
A. −2² is −4 (the square touches only the 2). The parentheses give 4 − 6 = −2, and 3 × (−2) = −6. Then −4 + (−6) = −10. −2 squares the minus sign too. The value is −10.
Evaluate: 3 × [12 − (5 + 1)] ÷ 2
Show the answer
B. Inside out: 5 + 1 = 6, then 12 − 6 = 6. Left to right: 3 × 6 = 18 and 18 ÷ 2 = 9. 12 drops the inner parentheses (12 − 5 + 1 = 8), and 18 forgets to divide. The value is 9.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
You finished order of operations.
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