MathsICSE Class 8 · Cubes and Cube Roots

Cube Root by Prime Factorisation: Triplets, Multipliers and Divisors (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

Factorise into primes and group equal primes in triplets. The cube root is the product of one prime from each triplet. Primes left outside a complete triplet show the least number to multiply by (to complete triplets) or to divide by (to remove them).

Group equal primes in triplets of three, then take one prime from each triplet and multiply them to get the cube root.

Illustration drawn for QuestionBuddy.

Understand Cube Root by Prime Factorisation

A perfect cube is the cube of a whole number: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \dots$ . The CISCE Class VIII curriculum asks for cube roots by the factor method only, for numbers of at most three digits. Several examples below (such as $1728$ and $3375$ ) are larger; they are an extension for olympiad-style practice and use the same method.

Method. Write the number as a product of primes. If each prime appears a multiple of three times, group them in triplets. The cube root takes one prime from each triplet. Example: $3375 = 3^{3} \times 5^{3}$ , so $33375 = 3 \times 5 = 15$ .

Negative numbers. The cube of a negative number is negative, so $3 - a = - 3 a$ . For instance, $3 - 4096 = - 16$ .

Fractions and decimals. $3 \frac{p}{q} = \frac{3 p}{3 q}$ . For decimals, shift the point by a multiple of three places: $0.003375 = \frac{3375}{1 0 ^{6}}$ , so the root is $\frac{15}{100} = 0.15$ .

Smallest multiplier. Multiply by the primes needed to bring each count up to a multiple of $3$ . If a prime appears twice, you need one more; if once, two more.

Smallest divisor. Divide by the primes that are left over after removing complete triplets.

Volume link. A cube of side $s$ has volume $s^{3}$ ; melting solids into one cube preserves total volume.

Cube Root by Prime Factorisation: formulas to remember

Cube root from primes: $3 p^{3 a} q^{3 b} = p^{a} q^{b}$; Divide every exponent by 3.
Negative cube roots: $3 - a = - 3 a$; Odd root keeps the sign.
Volume of a cube: $V = s^{3}$; Used for melting and recasting problems.

How to solve Cube Root by Prime Factorisation questions

Factorise, group in threes, then multiply, divide or take the root.

1Divide by primes in order to get the prime factorisation.
2Count each prime and group equal primes in triplets.
3If every count is a multiple of $3$ , take one prime per triplet for the root.
4Otherwise multiply by the missing primes, or divide by the leftover primes, as the question asks.
5Cube your root to verify.

Cube Root by Prime Factorisation in real life

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.

Recasting metal

Suppose three solid metal cubes of side 6 cm, 8 cm and 10 cm are melted together and recast as one cube (no metal is lost).

Find the side of the new cube.

Try it first, then show the solutionSolution

Total volume $= 216 + 512 + 1000 = 1728$ cm $^{3}$ .
$1728 = 2^{6} \times 3^{3}$ , so side $= 2^{2} \times 3 = 12$ cm.

Answer: 12 cm.

Memory hook: Volumes add; sides do not.

Tricks to remember Cube Root by Prime Factorisation

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

Triplets, not pairs
Squares need pairs of primes, cubes need triplets.
Use it when: Choosing the grouping.
Short of three
Count 1 needs two more; count 2 needs one more.
Use it when: Multiplier questions.

Cube Root by Prime Factorisation: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Direct cube root · 1 mark · MCQBoard level

$31728$ is:
(a) 8 (b) 12 (c) 14 (d) 24

Try it first, then show the solutionSolution

$1728 = 2^{6} \times 3^{3}$ , root $= 2^{2} \times 3 = 12$ .

Answer: (b)

Example 2: Cube root by factors · 2 marks · Short answerBoard level

Find $33375$ by the prime factor method.

Try it first, then show the solutionSolution

$3375 = 3 \times 1125 = 3^{2} \times 375 = 3^{3} \times 125 = 3^{3} \times 5^{3}$ .
Root $= 3 \times 5 = 15$ . Check: $1 5^{3} = 3375$ .

Answer: 15

Example 3: Smallest multiplier · 3 marks · Short answerHarder board level

Find the smallest number by which 1323 must be multiplied to give a perfect cube, and the cube root of the product.

Try it first, then show the solutionSolution

$1323 = 3^{3} \times 7^{2}$ .
$7$ appears twice, so one more $7$ is needed: multiply by $7$ .
Product $= 9261 = 3^{3} \cdot 7^{3}$ ; root $= 21$ . Check: $2 1^{3} = 9261$ .

Answer: Multiplier 7; cube root 21

Example 4: Smallest divisor · 3 marks · Short answerHarder board level

Find the smallest number by which 8640 must be divided to give a perfect cube, and the cube root of the quotient.

Try it first, then show the solutionSolution

$8640 = 2^{6} \times 3^{3} \times 5$ .
$5$ is left outside any triplet: divide by $5$ .
Quotient $= 1728 = 2^{6} \cdot 3^{3}$ , root $= 12$ .

Answer: Divisor 5; cube root 12

Example 5: Melting cubes · 4 marks · Case-basedHOTS / competency

Cubes of side 6 cm, 8 cm and 10 cm are melted and recast into a single cube. Find (i) its side and (ii) how many cm $^{2}$ of paper are needed to cover it completely.

Try it first, then show the solutionSolution

(i) Volume $= 6^{3} + 8^{3} + 1 0^{3} = 216 + 512 + 1000 = 1728$ ; side $= 12$ cm.
(ii) Total surface area $= 6 \times 1 2^{2} = 6 \times 144 = 864$ cm $^{2}$ .

Answer: (i) 12 cm; (ii) 864 cm $^{2}$

Where students lose marks in Cube Root by Prime Factorisation

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

Mistake: Pairing primes as for square roots
Cube roots need triplets.
Wrong working
$31728 = 2 \times 2 \times 3$ using pairs.
Right working
$2^{6} \cdot 3^{3}$ : triplets $2, 2, 2 ∣ 2, 2, 2 ∣ 3, 3, 3$ ; root $= 2 \cdot 2 \cdot 3 = 12$ .
Do this: Say "triplets" aloud when you group.
Usually costs: The answer.
Mistake: Multiplying by the leftover prime itself rather than what completes the triplet
A prime appearing twice needs one more, not two.
Wrong working
$1323 = 3^{3} \cdot 7^{2}$ : multiply by $49$ .
Right working
Multiply by $7$ to get $7^{3}$ .
Do this: Write the count and the shortfall for each prime.
Usually costs: The multiplier mark.

Exam tips

Memorise cubes up to $1 0^{3}$ for quick checks.
Show the triplet grouping in your working.
In recasting questions, equate volumes first.

Practice questions on Cube Root by Prime Factorisation

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 · MCQBoard level
Which of these is a perfect cube?
(a) 243 (b) 343 (c) 432 (d) 500
Hint
Look for an integer whose cube it is.
Answer
(b). $343 = 7^{3}$ .
Q2 · 2 marks · Short answerBoard level
Find $3 - 4096$ .
Hint
$4096 = 2^{12}$ .
Answer
$34096 = 2^{4} = 16$ , so $3 - 4096 = - 16$ .
Q3 · 3 marks · Short answerHarder board level
Find the smallest number by which 392 must be multiplied to get a perfect cube. Find the cube root of the product.
Hint
$392 = 8 \times 49$ .
Answer
$392 = 2^{3} \cdot 7^{2}$ ; multiply by $7$ : $2744 = 1 4^{3}$ . Root $14$ .
Q4 · 4 marks · Short answerHarder board level
Evaluate $3 \frac{- 343}{1728}$ and $3 0.003375$ .
Hint
For the decimal, write it as 3375 over 10^6.
Answer
$3 \frac{- 343}{1728} = \frac{- 7}{12}$ . $0.003375 = \frac{3375}{1 0 ^{6}}$ and root $= \frac{15}{100} = 0.15$ .
Q5 · 4 marks · Case-basedHOTS / competency
The volume of a cube is 74088 cm $^{3}$ . Find its side and its total surface area.
Hint
Side is the cube root; TSA = 6 x side squared.
Answer
$74088 = 2^{3} \cdot 3^{3} \cdot 7^{3}$ , side $= 42$ cm. TSA $= 6 \times 4 2^{2} = 6 \times 1764 = 10584$ cm $^{2}$ .

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 the cube root of -64?Show answer
-4.
2.How many 2s does 1728 contain?Show answer
Six (1728 = 2^6 x 3^3).
3.Cube of 15?Show answer
3375.
4.Smallest number to multiply 108 by for a perfect cube?Show answer
$108 = 2^{2} \cdot 3^{3}$ ; multiply by 2 to get 216.

Cube Root by Prime Factorisation: Triplets, Multipliers and Divisors (ICSE Class 8)

Understand Cube Root by Prime Factorisation

Cube Root by Prime Factorisation: formulas to remember

How to solve Cube Root by Prime Factorisation questions

Cube Root by Prime Factorisation in real life

Tricks to remember Cube Root by Prime Factorisation

Cube Root by Prime Factorisation: solved examples

Where students lose marks in Cube Root by Prime Factorisation

Exam tips

Practice questions on Cube Root by Prime Factorisation

Test your recall

Cube Root by Prime Factorisation: frequently asked questions

Can a negative number have a cube root?

Is every perfect cube also a perfect square?