MathsCBSE Class 8 · A Square and A Cube

Perfect Cubes and Cube Roots for Class 8

Q: Do negative numbers have cube roots?

Yes. The cube of -6 is -216, so the cube root of -216 is -6. This differs from square roots, which exist only for non-negative numbers in this chapter.

Q: Is every cube a perfect square?

No. 8 is a cube but not a square. A number that is both needs every exponent to be a multiple of 6, such as 64 and 729.

Q: How do I find a cube root of a decimal or fraction?

Take the cube root of the numerator and denominator separately, or of the number without its decimal point, then restore the decimal. For example the cube root of 0.027 is 0.3 since 27 is 3 cubed and 0.027 has three decimal places.

Q: Where does the word root come from?

The NCERT history note says the Sanskrit word mula, meaning root of a plant or origin, was used for taking roots (varga-mula for square root and ghana-mula for cube root).

Updated 17 October 2026 · Fact-checked

A perfect cube is the cube of a natural number: 1, 8, 27, 64, 125, ... A number is a perfect cube when its prime factors can be split into three identical groups, and the cube root is the product of one group. Class 8 questions test the smallest multiplier or divisor that gives a cube, cube roots by unit-digit estimation, and patterns such as taxicab numbers.

Group equal prime factors into triplets, then multiply one prime from each triplet: 3 × 5 = 15 is the cube root of 3375.

Illustration drawn for QuestionBuddy.

Understand Perfect Cubes and Cube Roots

A cube number is a number multiplied by itself three times: $n^{3} = n \times n \times n$ . It is the volume in unit cubes of a cube with edge $n$ units: a cube of edge 4 has 4 layers of $4 \times 4$ unit cubes, so $4^{3} = 64$ . The cubes of natural numbers (1, 8, 27, 64, 125, ...) are perfect cubes. You can also cube fractions, decimals and negatives, for example $(\frac{4}{6})^{3} = \frac{64}{216}$ and $(- 6)^{3} = - 216$ .

The cube root. If $y = x^{3}$ then $x$ is the cube root of $y$ , written $x = 3 y$ . So $38 = 2$ and $3 n^{3} = n$ .

Cubes and prime factors. When you cube a number, each prime factor appears three times as often: $12 = 2 \times 2 \times 3$ and $1 2^{3} = 1728 = 2^{6} \times 3^{3}$ . Therefore:

A number is a perfect cube exactly when every prime in its factorisation has an exponent that is a multiple of 3.
To find the cube root, split the prime factors into three equal groups (triplets) and multiply one from each triplet: $3375 = 3^{3} \times 5^{3}$ so $33375 = 3 \times 5 = 15$ .
$500 = 2^{2} \times 5^{3}$ is not a cube because 2 appears only twice.
To make a number a cube, multiply by the primes needed to complete each triplet, or divide away the extra primes.

Facts about cubes.

Unlike squares, cubes can end in any digit. The last digit of $n^{3}$ follows the last digit of $n$ : 0 to 0, 1 to 1, 2 to 8, 3 to 7, 4 to 4, 5 to 5, 6 to 6, 7 to 3, 8 to 2, 9 to 9.
A cube cannot end in exactly two zeros: if $n$ ends in $k$ zeros, $n^{3}$ ends in $3 k$ zeros, and $3 k = 2$ has no whole-number solution.
The cube of an odd number is odd and the cube of an even number is even. Cubes of 2-digit numbers have 4, 5 or 6 digits.
Cubes do not generally have an odd number of factors: 8 has four factors (1, 2, 4, 8).
Each cube is a sum of consecutive odd numbers: $2^{3} = 3 + 5$ , $3^{3} = 7 + 9 + 11$ , $4^{3} = 13 + 15 + 17 + 19$ . The cube $n^{3}$ uses $n$ odd numbers starting from $n^{2} - n + 1$ . So $1 0^{3} = 91 + 93 + \dots + 109$ .
Successive differences of the cubes 1, 8, 27, 64, 125, 216 settle to the constant 6 after three levels.

Estimating a cube root with no factorisation. Use the last digit of the cube to get the last digit of the root, and the number formed by dropping the last three digits to find which two cubes it lies between. This gives the tens digit.

Taxicab numbers. 1729 is the smallest number that is a sum of two positive cubes in two different ways: $1729 = 1^{3} + 1 2^{3} = 9^{3} + 1 0^{3}$ . In the NCERT story, Ramanujan pointed this out to Hardy. The next two such numbers are 4104 and 13832.

Squares and square roots are covered on the neighbouring topic page.

Perfect Cubes and Cube Roots: formulas to remember

Cube of a number: $n^{3} = n \times n \times n$; Volume of a cube of edge n.
Cube root: $3 n^{3} = n$; Cube root undoes cubing; it is defined for negatives too, e.g. cube root of -27 is -3.
Test for a cube: $every prime exponent is a multiple of 3$; Group the prime factors in triplets.
Cube as consecutive odd numbers: $first odd number = n^{2} - n + 1$; n cubed is the sum of n consecutive odd numbers beginning with this one (matches the NCERT pattern for n = 1 to 6 and 10).
Taxicab number 1729: $1729 = 1^{3} + 1 2^{3} = 9^{3} + 1 0^{3}$; Smallest sum of two cubes in two ways.

How to solve Perfect Cubes and Cube Roots questions

Follow these steps for any cube or cube-root question.

1Factorise into primes and write exponents.
2Group in triplets. Any prime with an exponent not divisible by 3 blocks the number from being a cube.
3Smallest multiplier: for each prime, add what is needed to reach the next multiple of 3. Smallest divisor: remove the remainder when the exponent is divided by 3.
4Cube root: take one prime from every triplet and multiply.
5For a large perfect cube with no factorisation: the last digit gives the root's last digit; bracket the thousands part between two cubes to get the tens digit.
6Verify by cubing your answer.

Perfect Cubes and Cube Roots 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.

Cube root from volume

Suppose a toy shop packs 1728 identical unit building blocks into a cube-shaped box with no gaps.

How many blocks lie along one edge of the box, and how many blocks form the bottom layer?

Try it first, then show the solutionSolution

Edge $= 31728$ . Now $1728 = 2^{6} \times 3^{3}$ , so the cube root is $2^{2} \times 3 = 12$ .
Bottom layer $= 12 \times 12 = 144$ blocks.

Answer: 12 blocks along each edge; 144 blocks in the bottom layer.

Memory hook: A cube box is built of equal layers: edge to the third power is layers times the area of one layer.

Melting and recasting

Imagine a workshop melts three solid metal cubes of edge 3 cm, 4 cm and 5 cm and casts the metal into a single cube, with no metal wasted.

Find the edge of the new cube.

Try it first, then show the solutionSolution

Total volume $= 27 + 64 + 125 = 216$ cm $^{3}$ .
Edge $= 3216 = 6$ cm.

Answer: 6 cm

Memory hook: Volume is conserved, so add the cubes, then take one cube root.

Tricks to remember Perfect Cubes and Cube Roots

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

Triplets for cubes, pairs for squares
Squares need pairs of prime factors, cubes need triplets. Count the matching group size.
Use it when: Choosing the right method in smallest-number problems.
Last digit swaps 2-8 and 3-7
Cubes keep 0, 1, 4, 5, 6, 9 unchanged. The pairs 2 and 8, and 3 and 7, swap: a root ending in 2 gives a cube ending in 8, and the reverse.
Use it when: Guessing cube roots like 4913 (ends in 3, so the root ends in 7).
Cubes of 1 to 10 as anchors
Learn cubes of 1 to 10: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. They fix the tens digit when you drop the last three digits of a big cube.
Use it when: Estimating cube roots like 17576: the part 17 lies between 8 and 27, so the tens digit is 2.
Ramanujan's 1729
Remember 1 and 12, 9 and 10. Their cubes add to 1729 in two different ways.
Use it when: Taxicab and number-pattern questions.

Perfect Cubes and Cube Roots: solved examples

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

Example 1: Reading a cube root · 1 mark · MCQBoard level

The value of $317576$ is: (a) 24 (b) 26 (c) 28 (d) 36

Try it first, then show the solutionSolution

17576 ends in 6, so the root ends in 6; only 26 and 36 qualify.
Drop the last three digits: 17. Since $2^{3} = 8 < 17 < 27 = 3^{3}$ , the tens digit is 2. So the root is 26.
Check: $2 6^{3} = 676 \times 26 = 17576$ .

Answer: (b) 26

Example 2: Smallest multiplier for a cube · 2 marks · Short answerBoard level

The number 4500 is multiplied by the least possible natural number so that the product is a perfect cube. What is the multiplier, and what is the cube root of the product?

Try it first, then show the solutionSolution

$4500 = 2^{2} \times 3^{2} \times 5^{3}$ . (1 mark)
2 and 3 each appear twice; they need one more each. Multiply by $2 \times 3 = 6$ : $27000 = 2^{3} \times 3^{3} \times 5^{3}$ and the cube root is $2 \times 3 \times 5 = 30$ . (1 mark)

Answer: Multiply by 6. The product 27000 has cube root 30.

Example 3: Smallest divisor for a cube · 2 marks · Short answerBoard level

Which least natural number, greater than 1, divides 5488 to leave a perfect cube as the quotient? Also state the cube root of that quotient.

Try it first, then show the solutionSolution

$5488 = 2^{4} \times 7^{3}$ . (1 mark)
2 appears 4 times, one more than a triplet, so divide by 2. Quotient $= 2744 = 2^{3} \times 7^{3}$ , with cube root $2 \times 7 = 14$ . (1 mark)

Answer: Divide by 2. The quotient 2744 has cube root 14.

Example 4: Cube root of a large number by estimation · 3 marks · Short answerHarder board level

Find $3658503$ without prime factorisation, given that it is a perfect cube.

Try it first, then show the solutionSolution

The last digit is 3. Cubes ending in 3 come from numbers ending in 7 (as $7^{3} = 343$ ). So the root ends in 7. (1 mark)
Drop the last three digits: 658. We have $8^{3} = 512 < 658 < 729 = 9^{3}$ , so the tens digit is 8. (1 mark)
The root is 87. Check: $8 7^{2} = 7569$ and $7569 \times 87 = 658503$ . (1 mark)

Answer: 87

Example 5: Least perfect cube divisible by three numbers · 3 marks · Short answerHarder board level

What is the least cube number that 6, 10 and 15 all divide exactly? Justify why no smaller cube works.

Try it first, then show the solutionSolution

LCM of $6 = 2 \times 3$ , $10 = 2 \times 5$ and $15 = 3 \times 5$ is $2 \times 3 \times 5 = 30$ . (1 mark)
Every common multiple is a multiple of 30. For a cube, each prime needs an exponent that is a multiple of 3, so we need $2^{3} \times 3^{3} \times 5^{3}$ . (1 mark)
This is $3 0^{3} = 27000$ . Check: $27000 \div 6 = 4500$ , $\div 10 = 2700$ , $\div 15 = 1800$ . (1 mark)

Answer: 27000

Where students lose marks in Perfect Cubes and Cube Roots

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

Mistake: Treating cubes like squares when grouping
Cubes need groups of three equal primes, not pairs.
Wrong working
"3375 = 3 x 3 x 3 x 5 x 5 x 5. Pair them: (3 x 3)(3)..."
Right working
$3375 = (3 \times 3 \times 3) (5 \times 5 \times 5) = 3^{3} \times 5^{3}$ , so the cube root is $3 \times 5 = 15$ .
Do this: Say the word "triplet" as you underline.
Usually costs: The method marks.
Mistake: Wrong last digit for the root
Mixing up the pairs 2-8 and 3-7.
Wrong working
"4913 ends in 3, so its cube root must end in 3."
Right working
$7^{3} = 343$ ends in 3, so the root of a cube ending in 3 ends in 7. $1 7^{3} = 4913$ .
Do this: Write the small table 1 to 9 cubed and read it backwards.
Usually costs: The answer and the check mark.
Mistake: Dividing when the question says multiply (or the reverse)
Multiplying and dividing give different smallest numbers for the same factorisation, so answering the wrong one loses the answer mark.
Wrong working
"Smallest number to divide 5488 = 2^4 x 7^3 by: multiply by 2^2 to complete the triplet, answer 4."
Right working
To divide, remove the extra 2: divide by 2 to get 2744. (To multiply instead, use $2^{2} = 4$ to get 21952.)
Do this: Underline multiply or divide in the question before starting.
Usually costs: All marks of the part.
Mistake: Claiming a cube has an odd number of factors
Only squares have an odd number of factors.
Wrong working
"8 is a cube, so it has an odd number of factors."
Right working
8 has factors 1, 2, 4, 8: four factors.
Do this: Link odd factor-count to squares only.
Usually costs: 1 mark in true-false items.

Exam tips

Write exponents in the factorisation line and mark the primes whose exponent is not a multiple of 3.
State whether you multiply or divide, and give the final cube and its root; many marks go to the root after the fix.
For estimated cube roots, show both the last-digit argument and the two bracketing cubes.
Memorise cubes of 1 to 10 and the cube digit table; it saves minutes in every paper.
In true-false answers, offer a counter-example such as 8 (four factors) or 2 cubed (even).

Practice questions on Perfect 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 · Assertion-reasonBoard level
Assertion (A): No perfect cube ends in exactly two zeros.
Reason (R): If a number ends in k zeros, its cube ends in 3k zeros.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Can 3k equal 2 for a whole number k?
Answer
(a). A cube $n^{3}$ ends in $3 k$ zeros, which is a multiple of 3, never 2. So A is true and R explains it.
Q2 · 3 marks · Short answerHarder board level
Express 12 cubed as the sum of 12 consecutive odd numbers. State the first and the last of them and verify the sum.
Hint
The first odd number in the sum for n cubed is n squared minus n plus 1.
Answer
First $= 1 2^{2} - 12 + 1 = 133$ . Twelve consecutive odds, so the last is $133 + 2 \times 11 = 155$ . Sum $= \frac{12 ( 133 + 155 )}{2} = 12 \times 144 = 1728 = 1 2^{3}$ . So $1 2^{3} = 133 + 135 + \dots + 155$ .
Q3 · 3 marks · Short answerHarder board level
How many digits can the cube of a two-digit number have? Justify with the smallest and largest cases and the boundaries.
Hint
Smallest is $1 0^{3}$ , largest is $9 9^{3}$ ; find where it moves from 4 to 5 and from 5 to 6 digits.
Answer
$1 0^{3} = 1000$ (4 digits) and $9 9^{3} = 970299$ (6 digits). 4 digits: $2 1^{3} = 9261$ ; 5 digits begin at $2 2^{3} = 10648$ ; $4 6^{3} = 97336$ is the last with 5 digits; $4 7^{3} = 103823$ has 6 digits. So the cube of a two-digit number has 4, 5 or 6 digits.
Q4 · 4 marks · Case-basedHOTS / competency
Case study: a metal workshop melts three solid cubes of edge 3 cm, 4 cm and 5 cm into one solid cube, with no loss of metal.
(i) Find the edge of the new cube. (1 mark)
(ii) Find the total surface area of the three original cubes and of the new cube. (2 marks)
(iii) By how many square centimetres does the surface area decrease? (1 mark)
Hint
Surface area of a cube of edge a is $6 a^{2}$ .
Answer
(i) Volume $= 27 + 64 + 125 = 216$ cm $^{3}$ , so edge $= 6$ cm. (ii) Original: $6 (9 + 16 + 25) = 6 \times 50 = 300$ cm $^{2}$ . New cube: $6 \times 36 = 216$ cm $^{2}$ . (iii) Decrease $= 300 - 216 = 84$ cm $^{2}$ .
Q5 · 4 marks · Long answerHOTS / competency
Compute the successive differences of the cubes 1, 8, 27, 64, 125, 216 until all differences at a level are equal. Without cubing 7, use your table to find 7 cubed and 8 cubed, and check them.
Hint
Level 3 is constant. Work backwards from it.
Answer
Level 1: 7, 19, 37, 61, 91. Level 2: 12, 18, 24, 30. Level 3: 6, 6, 6. Extend: next level 2 term $= 30 + 6 = 36$ ; next level 1 term $= 91 + 36 = 127$ ; next cube $= 216 + 127 = 343 = 7^{3}$ . Next: level 2 term $42$ , level 1 term $127 + 42 = 169$ , cube $343 + 169 = 512 = 8^{3}$ . Both agree with direct cubing: $7 \times 7 \times 7 = 343$ and $8 \times 8 \times 8 = 512$ .

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.How are the prime factors of a perfect cube grouped?Show answer
In triplets: every exponent is a multiple of 3.
2.What is the cube root of 3375?Show answer
15.
3.What is the smallest taxicab number and its two representations?Show answer
1729 = 1 cubed + 12 cubed = 9 cubed + 10 cubed.
4.Can a cube end in exactly two zeros?Show answer
No. Zeros at the end of a cube are a multiple of 3.
5.What last digit does the cube of a number ending in 7 have?Show answer
3.
6.How many consecutive odd numbers add up to n cubed?Show answer
n of them.

Perfect Cubes and Cube Roots: frequently asked questions

Do negative numbers have cube roots?

Yes. The cube of -6 is -216, so the cube root of -216 is -6. This differs from square roots, which exist only for non-negative numbers in this chapter.

Is every cube a perfect square?

No. 8 is a cube but not a square. A number that is both needs every exponent to be a multiple of 6, such as 64 and 729.

How do I find a cube root of a decimal or fraction?

Take the cube root of the numerator and denominator separately, or of the number without its decimal point, then restore the decimal. For example the cube root of 0.027 is 0.3 since 27 is 3 cubed and 0.027 has three decimal places.

Where does the word root come from?

The NCERT history note says the Sanskrit word mula, meaning root of a plant or origin, was used for taking roots (varga-mula for square root and ghana-mula for cube root).