MathsICSE Class 8 · Chapter 4

Cubes and Cube Roots ICSE Class 8: Perfect Cubes by the Prime Factor Method

Updated 15 October 2026 · Fact-checked

The cube of a number n is n × n × n, and the cube root of a number is the value whose cube gives that number. ICSE Class 8 finds cube roots only by the prime factor method: group equal primes in threes and take one prime from each group, for numbers of at most three digits.

What Cubes and Cube Roots is about

Multiplying a number by itself three times gives its cube: $4^{3} = 4 \times 4 \times 4 = 64$ . The reverse is the cube root: $364 = 4$ . Numbers such as 1, 8, 27, 64, 125, 216, 343, 512, 729 are perfect cubes.

The CISCE Class VIII curriculum asks for cubes and cube roots by the factor method only, for numbers containing at most 3 digits. So you should know the cubes of 1 to 10 by heart and be able to:

find cubes of integers, fractions and decimals,
decide whether a number is a perfect cube from its prime factors,
find the smallest number to multiply or divide by to get a perfect cube,
find cube roots of negative numbers, fractions and decimals built from 3-digit cubes (such as $3 - 0.729$ or $3 \frac{27}{125}$ ).

Unlike a square root, a cube root of a negative number exists: $3 - 8 = - 2$ , because $(- 2)^{3} = - 8$ . The name "cube" comes from volume: a cube of edge $a$ has volume $a^{3}$ .

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Cubes and Cube Roots sits in the Number System theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Cubes of integers, fractions and decimals

$a^{3} = a \times a \times a$ .

Negative number: three negative factors give a negative product, so $(- 7)^{3} = - 343$ .
Fraction: cube the top and the bottom separately: $(\frac{2}{5})^{3} = \frac{8}{125}$ . Change mixed numbers to improper fractions first.
Decimal: cube the digits and give the answer three times as many decimal places: $0. 3^{3} = 0.027$ (1 place becomes 3).

Learn the cubes of 1 to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Worked example

Find the cubes of $- 7$ , $\frac{2}{5}$ and $0.3$ .

Try it first, then show the solutionSolution

$(- 7)^{3} = - (7 \times 7 \times 7) = - 343$ .
$(\frac{2}{5})^{3} = \frac{2 ^{3}}{5 ^{3}} = \frac{8}{125}$ .
$3^{3} = 27$ and $0.3$ has 1 decimal place, so the cube has 3: $0.027$ .

Answer: $- 343$ , $\frac{8}{125}$ , $0.027$

Properties of cubes

The cube of an even number is even; the cube of an odd number is odd.
The cube of a negative number is negative.
The units digit of a cube depends only on the units digit of the number:

Units digit of n	0	1	2	3	4	5	6	7	8	9
Units digit of n³	0	1	8	7	4	5	6	3	2	9

Notice 2 and 8 swap, 3 and 7 swap, and the rest stay the same. Unlike squares, a cube can end in any digit.

Worked example

Without multiplying, find the units digit of $4 7^{3}$ .

Try it first, then show the solutionSolution

Only the units digit 7 matters.
$7^{3} = 343$ ends in 3.

Answer: 3

Testing for a perfect cube

Write the number as a product of primes and group equal primes in threes (triplets).

If every prime fits into a triplet, the number is a perfect cube.
If any prime is left over (alone or as a pair), it is not.

For example $216 = (2 \times 2 \times 2) (3 \times 3 \times 3)$ is a perfect cube, but $200 = (2 \times 2 \times 2) (5 \times 5)$ is not, because the two 5s do not make a triplet.

Worked example

Is 243 a perfect cube? If not, find the smallest number it must be multiplied by to make one.

Try it first, then show the solutionSolution

$243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^{5}$ .
One triplet $(3 \times 3 \times 3)$ and a pair $(3 \times 3)$ left over, so it is not a perfect cube.
The pair needs one more 3: $243 \times 3 = 729 = 3^{6} = (3^{2})^{3} = 9^{3}$ .

Answer: No; multiply by 3 to get $729 = 9^{3}$ .

Cube root by the prime factor method

Steps:
1. Find the prime factors of the number.
2. Make groups of three equal primes.
3. Take one prime from each group and multiply.

The product is the cube root. This is the only method required in Class VIII.

You can also use $3 ab = 3 a \times 3 b$ to split a product of two cubes.

Worked example

Find $3512$ by the prime factor method.

Try it first, then show the solutionSolution

$512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ (nine 2s).
Triplets: $(2 \times 2 \times 2) (2 \times 2 \times 2) (2 \times 2 \times 2)$ .
One from each: $2 \times 2 \times 2 = 8$ .

Answer: $8$

Cube roots of negatives, fractions and decimals

Negative: $3 - a = - 3 a$ . The sign comes out unchanged.
Fraction: $3 \frac{a}{b} = \frac{3 a}{3 b}$ . Change a mixed number to an improper fraction first: $3 3 \frac{3}{8} = 3 \frac{27}{8} = \frac{3}{2}$ .
Decimal: write it as a fraction with denominator 1000: $0.216 = \frac{216}{1000}$ , so $3 0.216 = \frac{6}{10} = 0.6$ . A perfect cube with 3 decimal places has a cube root with 1 decimal place.

Worked example

Find $3 \frac{- 343}{512}$ .

Try it first, then show the solutionSolution

Sign: the cube root of a negative number is negative.
$3343 = 7$ (as $343 = 7^{3}$ ) and $3512 = 8$ .
So the answer is $\frac{- 7}{8}$ .

Answer: $\frac{- 7}{8}$

Smallest multiplier or divisor for a perfect cube

Look at the prime factors that do not complete a triplet.

Divide by the leftover primes to remove them.
Multiply by whatever each leftover group needs to reach three: a single prime needs two more copies, a pair needs one more.

Then find the cube root from the triplets.

Worked example

Find the smallest number by which 192 must be divided so that the quotient is a perfect cube. Find the cube root of the quotient.

Try it first, then show the solutionSolution

$192 = 2^{6} \times 3 = (2 \times 2 \times 2) (2 \times 2 \times 2) \times 3$ .
The single 3 is left over, so divide by 3: $192 \div 3 = 64$ .
$364 = 2 \times 2 = 4$ .

Answer: Divide by 3; the cube root is 4.

Maths in real life: Cubes and Cube Roots

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Edge from volume

Suppose a cube-shaped gift box holds exactly 125 cm³.

How long is each edge of the box?

Try it first, then show the solutionSolution

Volume $= a^{3} = 125$ .
$125 = 5 \times 5 \times 5$ , so $a = 5$ cm.

Answer: 5 cm

Memory hook: Volume of a cube is edge cubed; the edge is the cube root.

Counting small cubes

Imagine you build one big solid cube from 343 sugar cubes, each of edge 1 cm.

How many sugar cubes lie along each edge?

Try it first, then show the solutionSolution

Cubes along an edge $= 3343$ .
$343 = 7^{3}$ , so 7 sugar cubes along each edge.

Answer: 7

Memory hook: n cubes along each edge makes n³ cubes in all.

Doubling the edge

Suppose a puzzle cube is made of 27 small cubes (3 along each edge), and a giant version has 6 small cubes along each edge.

How many times as many small cubes does the giant version have?

Try it first, then show the solutionSolution

Giant: $6^{3} = 216$ small cubes.
$216 \div 27 = 8$ , which is $2^{3}$ .

Answer: 8 times as many

Memory hook: Double the edge, and the volume becomes 2³ = 8 times.

Topics in this chapter

Cube Root by Prime Factorisation

Cubes and Cube Roots formula sheet

Cube: $a^{3} = a \times a \times a$; Volume of a cube of edge a.
Cube root: $3 a^{3} = a$; Works for negative a as well.
Negative number: $3 - a = - 3 a$; The sign comes out.
Product: $3 ab = 3 a \times 3 b$; Basis of the factor method.
Fraction: $3 \frac{a}{b} = \frac{3 a}{3 b}$; Use improper fractions.

How the school exam asks Cubes and Cube Roots

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Find cubes	Find the cube of −1.2, 2⅓ or 0.3.	Cube the digits; fix the sign; decimal places × 3.
Perfect cube test	Is 500 a perfect cube? Give reasons.	Prime factorise and look for triplets.
Cube root by factors	Find the cube root of 729 by the prime factor method.	Group in threes, take one from each group.
Smallest multiplier or divisor	Find the smallest number by which 108 must be multiplied (or 375 divided) to get a perfect cube.	Complete or remove the incomplete triplets.
Applications	Edge of a cube from its volume; numbers in a ratio with a given sum of cubes.	Volume = edge³; let the numbers be k, 2k, 3k and cube them.

Tricks to remember Cubes and Cube Roots

Short memory aids. Say each one aloud once, then use it on a question the same day.

Squares pair, cubes triple
Square root: groups of 2. Cube root: groups of 3. One member of each group goes into the root.
Use it when: Not mixing the two methods.
2–8 and 3–7 swap
In cubes, units digits 2 and 8 swap places, 3 and 7 swap places; 0, 1, 4, 5, 6, 9 stay as they are.
Use it when: Units digit of a cube, or of a cube root.
Sign survives
A cube keeps the sign of the number, and so does a cube root.
Use it when: Negative numbers.
Places × 3
Decimal places in the cube = 3 × places in the number.
Use it when: 0.3³ = 0.027; ∛0.216 = 0.6.

Cubes and Cube Roots: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Pick the perfect cube · 1 mark · MCQExam level

Which of these is a perfect cube?
(a) 200
(b) 216
(c) 256
(d) 324

Try it first, then show the solutionSolution

$216 = 2^{3} \times 3^{3} = 6^{3}$ .
$200 = 2^{3} \times 5^{2}$ , $256 = 2^{8}$ and $324 = 2^{2} \times 3^{4}$ all have incomplete triplets.

Answer: (b)

Example 2: Cube root by factors · 3 marks · Short answerExam level

Find $3729$ by the prime factor method.

Try it first, then show the solutionSolution

$729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3$ .
Triplets: $(3 \times 3 \times 3) (3 \times 3 \times 3)$ .
Root $= 3 \times 3 = 9$ .

Answer: $9$

Example 3: Cubes of a decimal and a mixed number · 3 marks · Short answerExam level

Find the cubes of (i) $- 1.2$ (ii) $2 \frac{1}{3}$ .

Try it first, then show the solutionSolution

(i) $1 2^{3} = 1728$ ; 1 decimal place becomes 3, and the sign stays negative: $- 1.728$ .
(ii) $2 \frac{1}{3} = \frac{7}{3}$ , so the cube is $\frac{343}{27}$ .
$\frac{343}{27} = 12 \frac{19}{27}$ .

Answer: (i) $- 1.728$ (ii) $\frac{343}{27} = 12 \frac{19}{27}$

Example 4: Smallest multiplier · 3 marks · Short answerHarder exam level

Find the smallest number by which 108 must be multiplied to get a perfect cube. Find the cube root of the product.

Try it first, then show the solutionSolution

$108 = 2 \times 2 \times 3 \times 3 \times 3$ .
The pair of 2s needs one more 2: $108 \times 2 = 216$ .
$216 = 2^{3} \times 3^{3}$ , so $3216 = 2 \times 3 = 6$ .

Answer: Multiply by 2; the cube root is 6.

Example 5: Fraction and decimal roots together · 4 marks · Long answerHarder exam level

Evaluate $3 \frac{- 512}{729} \times 3 0.125$ .

Try it first, then show the solutionSolution

$3512 = 8$ and $3729 = 9$ , so the first factor is $\frac{- 8}{9}$ .
$0.125 = \frac{125}{1000}$ , so $3 0.125 = \frac{5}{10} = \frac{1}{2}$ .
Product $= \frac{- 8}{9} \times \frac{1}{2}$ .
$= \frac{- 4}{9}$ .

Answer: $\frac{- 4}{9}$

Example 6: A cubical box · 4 marks · Long answerHarder exam level

A cubical box has a volume of 343 cm³. (i) Find the length of its edge. (ii) If the edge is doubled, find the new volume and how many times the old volume it is.

Try it first, then show the solutionSolution

(i) $343 = 7 \times 7 \times 7$ , so the edge is 7 cm.
(ii) New edge $= 14$ cm; new volume $= 1 4^{3} = 2744$ cm³.
$2744 \div 343 = 8$ .
So the volume becomes 8 times the old volume.

Answer: (i) 7 cm (ii) 2744 cm³, 8 times

Example 7: Numbers in a ratio · 5 marks · Long answerHOTS / competency

Three numbers are in the ratio $1 : 2 : 3$ , and the sum of their cubes is 288. Find the numbers.

Try it first, then show the solutionSolution

Let the numbers be $k$ , $2 k$ and $3 k$ .
Sum of cubes $= k^{3} + 8 k^{3} + 27 k^{3} = 36 k^{3}$ .
$36 k^{3} = 288$ , so $k^{3} = 8$ .
$8 = 2^{3}$ , so $k = 2$ .
The numbers are 2, 4 and 6. Check: $8 + 64 + 216 = 288$ ✓.

Answer: 2, 4 and 6

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

Packing cubes in a toy factory

Suppose a toy factory packs small wooden cubes of edge 2 cm into cube-shaped boxes, each with a volume of 512 cm³. The boxes are then packed into a cube-shaped carton that holds exactly 27 boxes.

(i) · 3 marks
Find the edge of a box by the prime factor method.
Answer
Answer: 8 cm
$512 = 2^{9} = (2^{3})^{3}$ , so the edge is $2^{3} = 8$ cm.
(ii) · 3 marks
How many wooden cubes fill one box, and how many lie along each edge?
Answer
Answer: 64 cubes; 4 along each edge
Volume of one wooden cube $= 2^{3} = 8$ cm³. Number $= 512 \div 8 = 64$ . $364 = 4$ along each edge (also $8 \div 2 = 4$ ).
(iii) · 4 marks
Find the edge and the volume of the carton.
Answer
Answer: 24 cm; 13824 cm³
27 boxes $= 3^{3}$ , so 3 boxes lie along each edge: edge $= 3 \times 8 = 24$ cm. Volume $= 27 \times 512 = 13824$ cm³ (check: $2 4^{3} = 13824$ ).

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Find $3216$ by the prime factor method. (3 marks)

Step	Marks
$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3$	1
Triplets $(2 \times 2 \times 2) (3 \times 3 \times 3)$	1
Root $= 2 \times 3 = 6$	1

Show that 243 is not a perfect cube. Find the smallest number by which it must be multiplied to make a perfect cube, and the cube root of the product. (4 marks)

Step	Marks
$243 = 3^{5}$ : one triplet and a pair	1
Pair left over, so not a perfect cube	1
Multiply by 3: $729 = 3^{6}$	1
$3729 = 3^{2} = 9$	1

Find the smallest number by which 256 must be (i) divided (ii) multiplied so that the result is a perfect cube. Find the cube root in each case. (5 marks)

Step	Marks
$256 = 2^{8}$ : two triplets and a pair of 2s	1
(i) Divide by $2^{2} = 4$ : $256 \div 4 = 64$	1
$364 = 4$	1
(ii) Multiply by 2: $256 \times 2 = 512 = 2^{9}$	1
$3512 = 2^{3} = 8$	1

Where students lose marks in Cubes and Cube Roots

The slips examiners see most often, and what to write instead.

Mistake: Pairing instead of tripling
Students use the square-root habit of pairs.
Wrong working
$3729$ : pairs $(3 \times 3) (3 \times 3) (3 \times 3)$ give 27
Right working
Triplets $(3 \times 3 \times 3) (3 \times 3 \times 3)$ give 9
Do this: For a cube root, circle groups of three.
Usually costs: The whole question.
Mistake: Wrong multiplier for a pair
A pair of primes needs only one more copy to make a triplet, and a single prime needs two more.
Wrong working
$108 = 2^{2} \times 3^{3}$ , so multiply by $2^{2} = 4$
Right working
Multiply by 2 to get $2^{3} \times 3^{3} = 216$
Do this: Count how many copies are missing from each incomplete group.
Usually costs: 2 to 3 marks.
Mistake: Dropping the minus sign
Cube roots of negative numbers exist and are negative.
Wrong working
$3 - 343 = 7$
Right working
$3 - 343 = - 7$
Do this: Deal with the sign first, then the number.
Usually costs: 1 mark.
Mistake: Wrong decimal places
Students give $0. 3^{3} = 0.27$ or $3 0.216 = 0.06$ .
Do this: Cube: places × 3. Cube root: places ÷ 3. Or convert the decimal to a fraction over 1000.
Usually costs: 1 mark.
Mistake: Not showing the factorisation
The syllabus method is the factor method, so the working (factors and groups) carries the marks.
Do this: Always show the prime factors and the groups before writing the root.
Usually costs: Up to 2 marks even when the answer is right.

Practice questions on Cubes and Cube Roots

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
The cube of a number ending in 8 ends in:
(a) 8
(b) 4
(c) 2
(d) 6
Hint
$8^{3} = 512$ .
Answer
(c)
Q2 · 2 marks · Short answerExam level
Find $3343$ by the prime factor method.
Hint
Divide by 7.
Answer
$343 = 7 \times 7 \times 7$ , so the root is 7.
Q3 · 2 marks · Short answerExam level
Is 500 a perfect cube? If not, find the smallest number by which it must be divided to make one.
Hint
$500 = 2^{2} \times 5^{3}$ .
Answer
No; the pair of 2s is incomplete. Divide by 4: $125 = 5^{3}$ .
Q4 · 3 marks · Short answerHarder exam level
Find the smallest number by which 72 must be multiplied to get a perfect cube, and the cube root of the product.
Hint
$72 = 2^{3} \times 3^{2}$ .
Answer
Multiply by 3: $216 = 2^{3} \times 3^{3}$ ; cube root 6.
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): The cube of every negative integer is negative.
Reason (R): The product of three negative numbers is negative.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
A cube is a product of three equal factors.
Answer
(c)
Q6 · 2 marks · Short answerHarder exam level
Find $3 3 \frac{3}{8}$ .
Hint
Improper fraction first.
Answer
$3 \frac{27}{8} = \frac{3}{2} = 1 \frac{1}{2}$ .
Q7 · 3 marks · Short answerHarder exam level
Find the smallest number by which 686 must be divided to get a perfect cube. Find the cube root of the quotient.
Hint
$686 = 2 \times 7^{3}$ .
Answer
Divide by 2: $343$ ; cube root 7.
Q8 · 2 marks · Short answerHarder exam level
Find $3 - 0.729$ .
Hint
$0.729 = \frac{729}{1000}$ .
Answer
$- \frac{9}{10} = - 0.9$ .
Q9 · 2 marks · Short answerHarder exam level
Using $3 ab = 3 a \times 3 b$ , find $3 125 \times 64$ .
Hint
Find each cube root separately.
Answer
$5 \times 4 = 20$ .
Q10 · 3 marks · Short answerHarder exam level
Find the smallest number by which 375 must be divided so that the quotient is a perfect cube. Find the cube root.
Hint
$375 = 3 \times 5^{3}$ .
Answer
Divide by 3: $125$ ; cube root 5.
Q11 · 3 marks · Short answerHOTS / competency
A solid metal cuboid measuring $12$ cm × $6$ cm × $3$ cm is melted and recast into a cube. Find the edge of the cube.
Hint
Volume stays the same.
Answer
Volume $= 12 \times 6 \times 3 = 216$ cm³ $= 2^{3} \times 3^{3}$ , so the edge is $6$ cm.
Q12 · 4 marks · Long answerHOTS / competency
The volumes of two cubes are 64 cm³ and 343 cm³. Find the ratio of (i) their edges (ii) the areas of one face of each.
Hint
Edges are the cube roots; face area = edge².
Answer
(i) Edges 4 cm and 7 cm, ratio $4 : 7$ . (ii) Face areas 16 cm² and 49 cm², ratio $16 : 49$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is 9³?Show answer
729.
2.What is ∛343?Show answer
7.
3.What is ∛(−125)?Show answer
−5.
4.What is the units digit of the cube of a number ending in 3?Show answer
7.
5.Is 100 a perfect cube?Show answer
No: 100 = 2² × 5².
6.What is ∛0.008?Show answer
0.2.
7.What is (−1/4)³?Show answer
−1/64.
8.Find ∛(8/27).Show answer
2/3.

Best order to study it

Learn the cubes of 1 to 10 and the units-digit table.
Practise cubes of negatives, fractions and decimals.
Learn the perfect-cube test by prime-factor triplets.
Find cube roots by the factor method, including negatives, fractions and decimals.
Finish with smallest multiplier or divisor and volume problems.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you list the cubes of 1 to 10 without hesitation?
Do you group prime factors in threes, not twos?
Do you know how many copies to add to complete a triplet?
Does your cube root of a negative number have a minus sign?
Do you convert decimals to fractions over 1000 for cube roots?
Can you get the edge of a cube from its volume?

Cubes and Cube Roots: frequently asked questions

Can a negative number have a cube root?

Yes. (−3)³ = −27, so ∛(−27) = −3. This is different from square roots, where a negative number has no square root among the numbers you know.

Do I need the long division or estimation method for cube roots?

No. The CISCE Class VIII curriculum asks only for the factor method, for numbers of at most three digits.

Can a perfect cube end in any digit?

Yes. Cubes can end in any of 0 to 9, which is why the units digit alone cannot rule out a cube the way it can rule out a square.

Why is a⁶ both a square and a cube?

a⁶ = (a³)² = (a²)³. For example 64 = 8² = 4³ and 729 = 27² = 9³.

How do I find the cube root of a mixed number?

Change it to an improper fraction, then find the cube roots of the numerator and denominator: ∛(3⅜) = ∛(27/8) = 3/2.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII