Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Number

Order of Operations and Calculating with Integers

Work out calculations with brackets, powers and roots in the correct order, and estimate, multiply and divide positive and negative integers.

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In the curriculum

Know that brackets, indices (powers, square roots and cube roots) and the four operations are always worked out in a fixed order (8Ni.01)
Estimate, multiply and divide positive and negative integers, and find and explain general rules (generalisations) about the signs of the answers (8Ni.02)

Is 2 × 3² equal to 36 or 18? Is (−6) × (−4) positive or negative? On this page you will learn the full order of operations for Stage 8 — now including powers and roots — and the rules for multiplying and dividing negative numbers, with quick ways to estimate answers so you can check your work. Everything on this page is Paper 1 (non-calculator) material, so practise it without a calculator.

Key words

Word / phrase	Meaning / use
integer	عدد صحيح — a whole number: positive, negative or zero (…, −2, −1, 0, 1, 2, …)
negative number	عدد سالب — a number less than zero, e.g. −7
index / power (plural: indices)	الأس — the small raised number: in 5³ the index is 3, so 5³ = 5 × 5 × 5
square root √	الجذر التربيعي — √49 = 7 because 7 × 7 = 49
cube root ∛	الجذر التكعيبي — ∛64 = 4 because 4 × 4 × 4 = 64
product	حاصل الضرب — the answer to a multiplication
quotient	خارج القسمة — the answer to a division
estimate	تقدير — round the numbers to easy values, then calculate, to get an approximate answer
generalisation	تعميم — a rule that is true for a whole group of numbers, not just one example
inverse operation	العملية العكسية — the operation that undoes another: × and ÷ are inverses

1. The order of operations — now with powers and roots

At Stage 8 the order of operations has four levels. Work from level 1 down to level 4. Inside one level, go from left to right.

The four levels of the order of operations. Levels 3 and 4 each contain two operations of equal rank.

Brackets. Work out everything inside brackets first. If there are brackets inside brackets, start with the innermost pair.
Indices (powers and roots). Work out squares, cubes, square roots and cube roots.
Multiplication and division, from left to right.
Addition and subtraction, from left to right.

Calculation	Working	Answer
3 + 4²	power first: 4² = 16, then 3 + 16	19
(3 + 4)²	bracket first: 7, then 7²	49
2 × 3²	power first: 3² = 9, then 2 × 9	18 (not 36)
(2 × 3)²	bracket first: 6, then 6²	36
50 − 2 × √49	root: √49 = 7; multiply: 2 × 7 = 14; then 50 − 14	36
∛64 + 6² ÷ 4	∛64 = 4 and 6² = 36; divide: 36 ÷ 4 = 9; then 4 + 9	13
5 × (8 − 2³)	inside the bracket the power comes first: 8 − 8 = 0; then 5 × 0	0
2 × (15 − (4 + 3))	innermost bracket: 4 + 3 = 7; then 15 − 7 = 8; then 2 × 8	16

Hidden brackets: root signs and fraction lines

Two symbols act like brackets even though you cannot see the brackets:

A root sign covers everything under its line. √(9 + 16) means: add first, 9 + 16 = 25, then √25 = 5. This is not the same as √9 + √16 = 3 + 4 = 7.
A fraction line groups the top and the bottom. In (12 + 8) ÷ (2 × 5), written as a fraction with 12 + 8 on top and 2 × 5 underneath, work out the top (20) and the bottom (10) first, then divide: 20 ÷ 10 = 2.

−3² is not the same as (−3)²

In −3² the power is done before the minus sign is applied, so −3² = −(3 × 3) = −9. In (−3)² the bracket tells you to square the negative number: (−3) × (−3) = 9. A scientific calculator follows the same rule, so type the brackets when you mean (−3)².

BIDMAS and your calculator

Many books use BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Remember that D and M have the same rank, and so do A and S — go left to right. A scientific calculator (allowed on Paper 2 only) follows the order of operations automatically, but only if you type the brackets you need. For a fraction such as (12 + 8) ÷ (2 × 5), type the brackets round the top and the bottom, or use the fraction key.

ترتيب العمليات في Stage 8

١- الأقواس الأول (ولو فيه قوس جوه قوس ابدأ بالداخلي).
٢- الأسس والجذور: التربيع والتكعيب والجذر التربيعي والتكعيبي.
٣- الضرب والقسمة — بالترتيب من الشمال لليمين.
٤- الجمع والطرح — برضه من الشمال لليمين.
خلي بالك: علامة الجذر وخط الكسر شغالين زي القوس. و −3² = −9 لكن (−3)² = 9.

2. Multiplying and dividing negative integers

Multiplication is repeated addition. 4 × (−3) means four lots of −3: start at 0 and make four jumps of −3. You land on −12.

4 × (−3) = −12. Because multiplication is commutative, (−3) × 4 = −12 too.

Why is negative × negative positive? Look at the pattern below (this is generalising: we look at many examples and find the rule). Each time the multiplier goes down by 1, the answer goes up by 3. Keep the pattern going past zero and the answers become positive.

Continuing the pattern shows that (−3) × (−1) = 3, (−3) × (−2) = 6 and (−3) × (−3) = 9.

This gives the sign rules. They work for multiplication and division, because division is the inverse of multiplication.

The sign rules for × and ÷.

Signs	Multiplication	Division	Sign of the answer
positive and positive	6 × 4 = 24	24 ÷ 4 = 6	positive
positive and negative	6 × (−4) = −24	24 ÷ (−4) = −6	negative
negative and positive	(−6) × 4 = −24	(−24) ÷ 4 = −6	negative
negative and negative	(−6) × (−4) = 24	(−24) ÷ (−4) = 6	positive

Fact families help with division. Because (−6) × 4 = −24, you also know (−24) ÷ 4 = −6 and (−24) ÷ (−6) = 4. If you are unsure of a division, turn it into a multiplication: (−56) ÷ (−8) = ? asks "what times −8 makes −56?" The answer is 7.

3. Generalisations about signs

Checkpoint likes to ask you to test a statement or explain a rule. These generalisations are always true:

Same signs give a positive answer; different signs give a negative answer (for × and ÷).
Count the negative signs when you multiply several numbers. An even number of negatives gives a positive answer; an odd number gives a negative answer. (−2) × (−3) × (−5) = 6 × (−5) = −30 (three negatives). (−1) × (−2) × (−3) × (−4) = 24 (four negatives).
Multiplying by −1 changes the sign: (−1) × 9 = −9 and (−1) × (−9) = 9.
The square of any non-zero integer is positive, because it is either positive × positive or negative × negative.
Zero: any number × 0 = 0, and 0 ÷ any non-zero number = 0. You can never divide by zero.
If a × b is negative, exactly one of a and b is negative. If a × b is positive (and not zero), a and b have the same sign.

To show a statement is false, one counter-example is enough (this is convincing). "The product of two integers is always bigger than both of them" is false: 5 × (−2) = −10, which is smaller than both 5 and −2.

4. Multiplying and dividing larger integers

Decide the sign first using the sign rules.
Ignore the signs and calculate with the positive numbers (grid method, column method or short division).
Put the sign on the answer, then check with an estimate.

Example: (−23) × 14. Different signs, so the answer is negative. Now work out 23 × 14 with a grid:

×	10	4	Row total
20	200	80	280
3	30	12	42
		Total	280 + 42 = 322

So (−23) × 14 = −322. Estimate to check: −20 × 15 = −300, which is close. ✓

Example: (−432) ÷ (−18). Same signs, so the answer is positive. 432 ÷ 18: 18 × 20 = 360, leaving 432 − 360 = 72, and 72 ÷ 18 = 4. So 432 ÷ 18 = 24 and (−432) ÷ (−18) = 24. Check: 18 × 24 = 432 ✓.

Example: 936 ÷ (−6). Different signs, so negative. Short division: 9 ÷ 6 = 1 remainder 3; 33 ÷ 6 = 5 remainder 3; 36 ÷ 6 = 6. So 936 ÷ 6 = 156 and 936 ÷ (−6) = −156.

5. Estimating

To estimate, round each number to one significant figure (or to another easy number), work out the easy calculation, and keep the sign from the sign rules. Estimates are used to check answers and to answer "Which is the best estimate…?" questions. Show the rounded numbers — the examiner wants to see them.

Calculation	Rounded	Estimate	Exact answer
(−48) × 31	(−50) × 30	−1500	−1488
612 ÷ (−19)	600 ÷ (−20)	−30	about −32.2
(−79) × (−52)	(−80) × (−50)	4000	4108
(−8 213) ÷ 41	(−8 000) ÷ 40	−200	−200.3…

Sometimes it is better to round to a number that divides easily rather than to exactly 1 significant figure. For 3 580 ÷ (−62), rounding to 3 600 ÷ (−60) gives a neat estimate of −60.

6. Putting it together: mixed calculations with negatives

Use the order of operations first, then the sign rules at each step. Remember that subtracting a negative is the same as adding: 5 − (−3) = 5 + 3 = 8.

Example: (−4) × 3 − (−20) ÷ 5. Do × and ÷ first: (−4) × 3 = −12 and (−20) ÷ 5 = −4. Then −12 − (−4) = −12 + 4 = −8.

Example: (−2)³ + 10 ÷ (−5). Power first: (−2)³ = (−2) × (−2) × (−2) = −8. Division: 10 ÷ (−5) = −2. Then −8 + (−2) = −10.

Worked examples

Worked example 1 — Work out 4 + 3 × 2³. [1 mark]

Power first: 2³ = 2 × 2 × 2 = 8.
Multiply: 3 × 8 = 24.
Add: 4 + 24 = 28. Answer: 28

Common mistake: doing 4 + 3 = 7 first and getting 7 × 8 = 56, or working out 3 × 2 = 6 first and then 6³.

Worked example 2 — Work out (a) √(100 − 36) (b) 100 − √36. [2 marks]

(a) The root sign covers 100 − 36, so subtract first: 100 − 36 = 64. Then √64 = 8.
(b) Only 36 is under the root sign: √36 = 6. Then 100 − 6 = 94.

Common mistake: in (a), writing √100 − √36 = 10 − 6 = 4. You cannot take the root of each part separately.

Worked example 3 — Work out (−6) × (−4) ÷ (−3). [1 mark]

× and ÷ have equal rank, so go left to right.
(−6) × (−4) = 24 (same signs → positive).
24 ÷ (−3) = −8 (different signs → negative). Answer: −8
Check by counting negatives: three negatives (odd) → negative answer ✓.

Worked example 4 — Kareem says, "When you multiply three negative numbers together, the answer is always positive." Is Kareem correct? Show how you know. [1 mark]

Test the statement with an example (specialising): (−1) × (−2) × (−3).
(−1) × (−2) = 2, then 2 × (−3) = −6.
The answer is negative, so Kareem is not correct. Three is an odd number of negatives, so the product of three negative numbers is always negative.

Common mistake: only writing "No". The mark is for the evidence — a correct counter-example or a correct reason.

Worked example 5 — (a) Estimate (−412) × 29. Show your rounding. (b) Use your estimate to choose the exact answer: −1 948, −11 948, 11 948, −119 480. [2 marks]

(a) Round: −412 ≈ −400 and 29 ≈ 30. Estimate: (−400) × 30 = −12 000.
(b) The answer must be negative and close to −12 000. Only −11 948 fits.
(Check: 412 × 29 = 412 × 30 − 412 = 12 360 − 412 = 11 948.)

Common mistake: forgetting the negative sign in the estimate, or rounding 412 to 410 and 29 to 29 — that is not an estimate, it is still a hard calculation.

Worked example 6 — Put one pair of brackets into this calculation to make it correct: 3 + 5² − 4 × 2 = 56. [1 mark]

Without brackets: 3 + 25 − 8 = 20. We need a much bigger answer.
56 is close to 64 = 8², and 64 − 8 = 56. And 8 = 3 + 5.
Try (3 + 5)² − 4 × 2 = 8² − 8 = 64 − 8 = 56 ✓
Answer: (3 + 5)² − 4 × 2 = 56

Common mistake: putting the bracket round 3 + 5²: (3 + 5²) − 4 × 2 = 28 − 8 = 20. The bracket must make the square apply to 3 + 5.

Common mistakes

Squaring before working out the bracket, or squaring only one number: 2 × 3² = 18, but (2 × 3)² = 36.
Splitting a root: √(9 + 16) = 5, not √9 + √16 = 7.
Writing −3² = 9. Without brackets, −3² = −9; only (−3)² = 9.
Doing × before ÷ (or + before −) instead of going left to right: 24 ÷ 4 × 2 = 12, not 3.
Mixing up the rules for adding and multiplying negatives: (−5) + (−3) = −8, but (−5) × (−3) = +15.
Losing the sign in a long calculation. Decide the sign first and write it down before you calculate.
Rounding badly when estimating: round to 1 significant figure (or an easy number) and show the rounded numbers.

How Checkpoint asks this

"Work out" a mixed calculation with brackets, powers or roots — usually on Paper 1 (no calculator), worth [1] or [2].
"Insert brackets to make the calculation correct" — try the brackets that make one operation happen earlier, and check each try fully.
"Draw a ring around" the calculation with a given answer, or the best estimate.
"Is she correct? Explain your answer" — give a calculation as evidence, e.g. a counter-example. Words alone rarely earn the mark.
"Write the missing number": (−6) × ☐ = 42. Use the inverse: 42 ÷ (−6) = −7.
On Paper 2 you may use a calculator, but you must still type brackets correctly — and an estimate tells you if you made a typing slip.

Practice

Remember

Order: Brackets → Indices (powers and roots) → × and ÷ left to right → + and − left to right.
A root sign and a fraction line act like brackets: √(9 + 16) = √25 = 5.
−3² = −9, but (−3)² = 9.
× and ÷: same signs → positive; different signs → negative. Count the negatives when you multiply several numbers.
For big numbers: decide the sign, calculate without signs, attach the sign, then check with an estimate (round to 1 significant figure).

All pages in this course

Start here

Stage 8 Maths: The Tests and How to Prepare

Number

Algebra

Geometry and measure

Statistics and probability

Practice tests