Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Number

Squares, Cubes and Roots (Positive and Negative)

Square and cube positive and negative numbers, find both square roots of a number and the single cube root, estimate roots, and use powers and roots in longer calculations.

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

In the curriculum

Recognise the squares of positive and negative numbers, for example (−6)² = 36, and the square roots that go with them, for example the square roots of 36 are 6 and −6 (8Ni.06)
Recognise positive and negative cube numbers, for example (−4)³ = −64, and their cube roots, for example ∛(−64) = −4 (8Ni.07)

In Stage 7 you learned the square numbers and the cube numbers made from positive whole numbers. In Stage 8 you go further: you square and cube negative numbers, you learn that a positive number has two square roots, and you see why a cube root can be negative. This page also shows you how to estimate roots that are not whole numbers, and how to use roots and powers inside longer calculations.

Key words

Word / phrase	Meaning / use
square (verb)	يربّع — multiply a number by itself: 5² = 5 × 5 = 25
square number	عدد مربع — the answer when a whole number is squared: 1, 4, 9, 16, 25 …
square root (√)	الجذر التربيعي — the number that was squared: the square roots of 25 are 5 and −5; √25 = 5
cube (verb)	يكعّب — multiply a number by itself three times: 4³ = 4 × 4 × 4 = 64
cube number	عدد مكعب — the answer when a whole number is cubed: 1, 8, 27, 64 … and −1, −8, −27 …
cube root (∛)	الجذر التكعيبي — the number that was cubed: ∛64 = 4 and ∛(−64) = −4
index / power	الأس — the small raised number: in 7³ the index 3 tells you how many 7s are multiplied
integer	عدد صحيح — a whole number: positive, negative or zero
inverse	العملية العكسية — the operation that undoes another: square root undoes squaring

1. Squares and cubes: the pictures behind the words

A square number is the area of a square whose sides are a whole number long. A cube number is the volume of a cube made from small unit cubes. That is why we say "squared" for the power 2 and "cubed" for the power 3.

4² = 4 × 4 = 16 small squares. 3³ = 3 × 3 × 3 = 27 small cubes.

You should know these by heart for the non-calculator paper. Learn the squares up to 15² and the cubes of 1 to 5 and 10.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

n	1	2	3	4	5	10
n³	1	8	27	64	125	1000
(−n)³	−1	−8	−27	−64	−125	−1000

Useful facts: 20² = 400, 30² = 900, 0² = 0, 0³ = 0, 1² = 1³ = 1. Numbers like 1 and 64 are both square and cube numbers (64 = 8² = 4³).

2. Squaring negative numbers

Squaring means multiplying a number by itself. When the number is negative, you multiply two negatives, and negative × negative = positive.

(−3)² = (−3) × (−3) = 9
(−8)² = (−8) × (−8) = 64
(−1)² = 1
(−12)² = 144

Rule: the square of any number that is not zero is positive. So 6² and (−6)² give the same answer, 36.

Brackets change everything: (−5)² is not −5²

(−5)² means (−5) × (−5) = 25. The bracket says: square the whole of −5.
−5² means −(5²) = −(5 × 5) = −25. Indices come before the minus sign in the order of operations, so only the 5 is squared.
On a calculator, type the bracket: ( (−) 5 ) x² gives 25. Without the bracket many calculators give −25.

3. Square roots: two answers, one symbol

Finding a square root is the inverse (opposite) of squaring. Ask: "Which number, multiplied by itself, gives this?" For 9 there are two answers, because 3 × 3 = 9 and (−3) × (−3) = 9.

Squaring sends 3 and −3 to the same answer, 9. So 9 has two square roots.

You see…	It means…	Answer
"the square roots of 49"	every number that squares to give 49	7 and −7 (written ±7)
√49	the positive square root only	7
−√49	the negative square root	−7
x² = 49, find x	all values of x	x = 7 or x = −7
√(−49)	a number that squares to give −49	none — a square is never negative

The sign ± is read "plus or minus". So ±7 means "7 or −7". Remember: a negative number has no square root in the numbers you use at Stage 8, because every square is positive or zero.

Square roots of bigger squares. Use what you know: 30² = 900, so √900 = 30. 400 = 20², so the square roots of 400 are ±20. For 1600, think 40 × 40 = 1600, so √1600 = 40.

4. Cubing negative numbers and cube roots

Cubing multiplies three copies of the number. With a negative number, the first two negatives make a positive, and the third makes it negative again:

(−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8
(−5)³ = (−5) × (−5) × (−5) = 25 × (−5) = −125
(−10)³ = −1000

Rule: the cube of a positive number is positive; the cube of a negative number is negative. Cubing keeps the sign.

Each cube number has exactly one cube root, with the same sign.

Because cubing keeps the sign, every number has exactly one cube root, and you can find the cube root of a negative number:

Cube root	Think	Answer
∛27	3 × 3 × 3 = 27	3
∛(−27)	(−3) × (−3) × (−3) = −27	−3
∛125	5³ = 125	5
∛(−1000)	(−10)³ = −1000	−10
∛(−1)	(−1)³ = −1	−1

	Square root	Cube root
of a positive number, e.g. 64	two roots: 8 and −8 (√64 = 8)	one root: ∛64 = 4
of zero	0	0
of a negative number, e.g. −64	none	one root: ∛(−64) = −4

افتكر القاعدة دي

المربع دايماً موجب: (−٣)² = ٩ زي ٣² = ٩. علشان كده العدد الموجب له جذرين تربيعيين: ٣ و −٣. لكن علامة √ لوحدها معناها الجذر الموجب بس: √٩ = ٣.
المكعب بيحافظ على الإشارة: (−٢)³ = −٨. علشان كده الجذر التكعيبي واحد بس وليه نفس الإشارة: ∛(−٨) = −٢.
خلي بالك من القوس: (−٥)² = ٢٥ لكن −٥² = −٢٥.

5. Estimating roots that are not whole numbers

√40 is not a whole number, because 40 is not a square number. You can still say which two whole numbers it lies between:

Find the square numbers either side of 40: 36 and 49.
Take their square roots: √36 = 6 and √49 = 7.
So 6 < √40 < 7.
40 is 4 away from 36 but 9 away from 49, so √40 is nearer to 6. (A calculator gives 6.324…)

√40 sits between 6 and 7, nearer to 6.

Cube roots work the same way with cube numbers: 27 < 50 < 64, so 3 < ∛50 < 4. For a negative number, −64 < −50 < −27, so −4 < ∛(−50) < −3.

6. Powers and roots inside longer calculations

Powers and roots come after brackets and before ×, ÷, + and −. A root sign that covers a whole expression acts like a bracket: work out everything under it first.

Calculation	Working	Answer
√(9 + 16)	√25	5
√9 + √16	3 + 4	7 (not the same as above!)
(−2)³ + 3²	−8 + 9	1
2 × (−4)²	2 × 16	32
(2 × (−4))²	(−8)²	64
∛(−8) × √25	−2 × 5	−10

Substituting negative numbers. When a letter stands for a negative number, put it in a bracket before you square or cube: if x = −3, then x² = (−3)² = 9 and x³ = (−3)³ = −27.

Worked examples

Worked example 1 — Work out (a) (−7)² (b) −7² (c) (−1)³ [3 marks]

(a) (−7) × (−7) = 49 (negative × negative = positive).
(b) Square first, then apply the minus: −(7 × 7) = −49.
(c) (−1) × (−1) × (−1) = 1 × (−1) = −1.

Common mistake: giving 49 for part (b). Without a bracket, only the 7 is squared.

Worked example 2 — (a) Write down the value of √121. (b) Write down all the solutions of x² = 121. [2 marks]

(a) 11 × 11 = 121, and √ means the positive root: √121 = 11.
(b) Both 11 and −11 square to 121: x = 11 or x = −11 (x = ±11).

Common mistake: giving only 11 in part (b). "Solutions" or "all the values" means give both.

Worked example 3 — Work out ∛(−216). [1 mark]

Find the cube root of 216 first by trying cubes: 5³ = 125 (too small), 6³ = 6 × 6 × 6 = 36 × 6 = 216 ✓. The number is negative, and cubing keeps the sign, so ∛(−216) = −6. Check: (−6) × (−6) × (−6) = 36 × (−6) = −216 ✓.

Worked example 4 — Ring the two whole numbers that √90 lies between: 8 and 9 / 9 and 10 / 10 and 11 / 44 and 46 [1 mark]

Square numbers near 90: 81 = 9² and 100 = 10². 81 < 90 < 100, so 9 < √90 < 10: ring "9 and 10". Common mistake: halving 90 and choosing "44 and 46" — a square root is not half of the number.

Worked example 5 — Sami says, "√(25 − 16) is the same as √25 − √16." Is he correct? Show how you decide. [2 marks]

√(25 − 16) = √9 = 3 (work under the root first).
√25 − √16 = 5 − 4 = 1.
3 ≠ 1, so Sami is not correct. (1 mark for each value, or for one value and a correct conclusion.)

Worked example 6 — Find the value of 2a² − a³ when a = −3. [2 marks]

Put a = −3 in brackets: 2 × (−3)² − (−3)³.
Powers first: (−3)² = 9 and (−3)³ = −27.
2 × 9 − (−27) = 18 + 27 = 45.

Common mistake: writing −3² = −9 instead of (−3)² = 9, or forgetting that subtracting −27 means adding 27.

Common mistakes

Thinking 5² = 10 or 4³ = 12. Squaring is 5 × 5, not 5 × 2; cubing is 4 × 4 × 4, not 4 × 3.
Writing −6² = 36. Without a bracket only the 6 is squared: −6² = −36, but (−6)² = 36.
Giving only the positive answer when the question asks for "the square roots" or "all solutions" of x² = 64. Give ±8.
Writing √25 = ±5. The symbol √ on its own means the positive root: √25 = 5.
Saying ∛(−8) is impossible. It is −2, because (−2)³ = −8. It is the square root of a negative number that does not exist.
Saying (−2)³ = 8. Count the negatives: three negatives multiply to a negative.
Splitting a root over + or −: √(9 + 16) is 5, not 3 + 4 = 7.

How Checkpoint asks this

Short "work out" questions with negative numbers, e.g. (−4)² or ∛(−125), usually worth 1 mark each. These are on the non-calculator paper, so know the squares and cubes in the tables above.
"Draw a ring around" the correct value or the two numbers a root lies between.
"Tick (✓) Correct or Not correct" for a student's statement, then explain. Give a calculation as your reason, e.g. "(−3)² = 9, not −9".
"Write down all the solutions" or "Find both values of x" — this is the signal to give the positive and negative square root.
Substitution: "Find the value of … when x = −2." Write the bracket (−2) every time.
Investigation-style (Thinking and Working Mathematically): "Give an example to show that … is false" — one correct counter-example is enough, e.g. 0.5² = 0.25 is smaller than 0.5.

إزاي تجاوب في الامتحان

لو السؤال قال «كل الحلول» أو «الجذرين التربيعيين» اكتب الإجابة الموجبة والسالبة: ±٨.
لو لقيت √ لوحدها اكتب الجذر الموجب بس.
عوّض بالعدد السالب جوه قوس دايماً: (−٢)² مش −٢².

Practice

Remember

Squaring a non-zero number always gives a positive answer: (−6)² = 6² = 36.
A positive number has two square roots (±); √ on its own means the positive one. Negative numbers have no square root.
Cubing keeps the sign: (−4)³ = −64, so ∛(−64) = −4. Every number has exactly one cube root.
Brackets matter: (−5)² = 25 but −5² = −25. Put negative numbers in brackets when you substitute.
To estimate a root, find the square (or cube) numbers on either side: 36 < 40 < 49 so 6 < √40 < 7.

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