TACHS: Order of operations: exponents before products before sums
Evaluate expressions with exponents, products and negative bases in the right order, and spot the step where a wrong answer went off track.
In what order do you work out an expression that has exponents, multiplication and addition?
Short answer
Parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. An exponent means repeated multiplication of the base.
Know first: Multiplication facts, Adding and subtracting negative numbers
Step 1 of 6
Exponents come before multiplication, which comes before addition
An exponent tells you how many times to multiply the base by itself: 2³ = 2 × 2 × 2 = 8. It never means base × exponent, so 2³ is not 6.
In any expression, work in this order. Rewrite the whole expression after each step so no term gets lost.
| Step | What to do | Example |
|---|---|---|
| 1. Parentheses | Work inside first | (6 − 2)² becomes 4² |
| 2. Exponents | Write out the repeated multiplication | 4² = 4 × 4 = 16 |
| 3. Multiply and divide | Left to right, as they appear | 18 ÷ 3 × 2 = 6 × 2 = 12 |
| 4. Add and subtract | Left to right, as they appear | 16 ÷ 8 + 1 = 2 + 1 = 3 |
Multiplication does not always come before division; whichever appears first, left to right, goes first.
Step 2 of 6
One operation at a time
Worked example: An exponent, a product and two sums
Find the value of 5 + 2³ × 3 − 4.
- A19
- B25
- C29
- D35
- 1
Exponent first
2³ = 2 × 2 × 2 = 8. Now 5 + 8 × 3 − 4.
- 2
Multiply next
8 × 3 = 24. Now 5 + 24 − 4.
- 3
Add and subtract left to right
5 + 24 = 29, then 29 − 4 = 25.
- 4
See why the other choices are there
19 reads 2³ as 2 × 3 = 6. 29 stops before subtracting 4. 35 adds before multiplying: (5 + 8) × 3 − 4.
Answer
The value is 25.
A student's work
A student finds the value of 10 − 2 × 3².
- 1
Exponent first: 3² = 9, so the expression is 10 − 2 × 9.
- 2
Subtract next: 10 − 2 = 8, so the expression is 8 × 9.
What went wrong
Multiplication comes before subtraction. The 2 belongs to 2 × 9, not to 10 − 2.
- 3
Multiply: 8 × 9 = 72.
The fix
After the exponent, multiply: 2 × 9 = 18. Then subtract: 10 − 18 = −8.
Step 3 of 6
Parentheses decide what a negative base means
A negative base in parentheses is multiplied with its sign: (−4)² = (−4) × (−4) = 16. Without parentheses, the exponent applies only to the 4: −4² = −(4 × 4) = −16.
An even number of negative factors gives a positive result, and an odd number gives a negative one. So (−2)³ = −8 and (−2)⁴ = 16.
Two powers, one subtraction
(−2)⁴ − 3²
Exponents: (−2)⁴ = 16 and 3² = 9
Subtract: 16 − 9 = 7
Step 4 of 6
Slow down at every exponent
Check yourself · Question 1
What is the value of 20 − 3 × 2³?
Answer: B
Exponent: 2³ = 8. Multiply: 3 × 8 = 24. Subtract: 20 − 24 = −4.
Trap answer: 136
Working strictly left to right
- Why it looks right
- Reading left to right is how you read words, so 20 − 3 looks like the first thing to do.
- Why it's wrong
- Multiplication comes before subtraction no matter where it sits. The 3 is multiplied by 8 before anything is subtracted from 20.
- The right answer
- −4, because 20 − 3 × 8 = 20 − 24 = −4.
Check yourself · Question 2
What is the value of (−5)² − 2⁴?
Answer: C
(−5)² = (−5) × (−5) = 25. 2⁴ = 2 × 2 × 2 × 2 = 16. Then 25 − 16 = 9.
Check yourself · Question 3
Which expression has the same value as 2 + 4 × 3²?
Answer: A
2 + 4 × 3² = 2 + 4 × 9 = 2 + 36 = 38, and 2 × 19 = 38.
Common mistake
"5² is 5 times 2, so it's 10."
- Why it's tempting
- The small 2 sits right next to the 5, and multiplying them is the quickest thing to do.
- Do this instead
- Write the power out before you use it: 5² = 5 × 5 = 25. The exponent counts how many 5s are multiplied.
Step 5 of 6
What to remember
Remember
- Parentheses, then exponents, then multiply and divide left to right, then add and subtract left to right.
- An exponent counts repeated factors: 2³ = 2 × 2 × 2 = 8.
- (−4)² = 16 but −4² = −16. Parentheses decide whether the sign is squared.
Step 6 of 6
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, what is the square written as a product, and which wrong choice comes from reading it as base × exponent?