Cube and Cube Root: Formulas, Tricks and Methods

Cube and Cube Root covers the third power of a number, methods to find cube roots, perfect cube identification and useful algebraic identities. This page explains cube formulas, unit-digit rules, prime factorisation and the grouping method for solving cube root questions quickly and accurately.

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What Are Cube and Cube Root?

The cube of a number n is n³ = n × n × n. The cube root of a number N is the value n such that n³ = N, written as ∛N = n.

For example, 5³ = 5 × 5 × 5 = 125, so ∛125 = 5. Cubes of positive numbers are positive, cubes of negative numbers are negative, and 0³ = 0. A number is a perfect cube if its cube root is an integer.

Cube and Cube Root Formula & Tricks

Important Formulas

Cube of a sum
(a + b)³ = a³ + 3a²b + 3ab² + b³

Use this identity when a number can be expressed as the sum of two simpler terms.

Cube of a difference
(a − b)³ = a³ − 3a²b + 3ab² − b³

The signs alternate when a difference is raised to the third power.

Sum of two cubes
a³ + b³ = (a + b)(a² − ab + b²)

This factorises an expression containing the sum of two cubes.

Difference of two cubes
a³ − b³ = (a − b)(a² + ab + b²)

This factorises an expression containing the difference of two cubes.

Cube root by prime factors
If N = p₁³ᵃ × p₂³ᵇ × ..., then ∛N = p₁ᵃ × p₂ᵇ × ...

Prime factors are grouped in sets of three equal factors. One factor from each group is taken outside the cube root.

Quick Tricks

Use the last digit to find the unit digit of a cube root

The unit digit of a perfect cube determines the unit digit of its cube root. The mapping is 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2 and 9→9.

Example: The number 12167 ends in 7, so its cube root ends in 3. Since 2³ = 8 and 3³ = 27, the first group 12 gives the tens digit 2. Therefore, ∛12167 = 23.
Separate a perfect cube into three-digit groups

Starting from the right, place commas or bars after every three digits. The largest cube less than or equal to the first group gives the leading digit of the cube root. The final digit gives its unit digit.

Example: For 250047, write 250|047. The largest cube not exceeding 250 is 6³ = 216, and the last digit 7 gives a root ending in 3. Hence, ∛250047 = 63.
Use nearby cubes for estimation

If a number is not a perfect cube, locate it between two consecutive cubes. Its cube root lies between the corresponding consecutive integers.

Example: Since 7³ = 343 and 8³ = 512, ∛400 lies between 7 and 8.

Cube and Cube Root Concepts

Perfect Cube Test Using Prime Factorisation

A positive integer is a perfect cube if the exponent of every prime factor in its prime factorisation is a multiple of 3.

For N = 2ᵃ × 3ᵇ × 5ᶜ × ..., N is a perfect cube only when a, b, c and all other exponents are divisible by 3. For example, 216 = 2³ × 3³ = (2 × 3)³, so it is a perfect cube.

Example: 1728 = 2⁶ × 3³ = (2² × 3)³ = 12³. Therefore, ∛1728 = 12.

Cube Root by Prime Factorisation

To find a cube root by prime factorisation, divide the number into groups of three identical prime factors and select one factor from each group.

Factorise the number completely, arrange equal prime factors together, and take one factor from every group of three outside the root sign. If any prime factor cannot be grouped in threes, the number is not a perfect cube.

Example: ∛3375 = ∛(3⁳ × 5⁳) = 3 × 5 = 15.

Cube Root by Digit Grouping

For a perfect cube, divide the digits into groups of three from the right to identify its cube root.

The first group determines the leading digit or digits. Find the greatest integer whose cube is at most the first group. The last digit of the number determines the unit digit using the cube unit-digit mapping. This method is especially useful for large perfect cubes.

Example: For 389017, write 389|017. Since 7³ = 343 and 8³ = 512, the first root digit is 7. The last digit 7 requires a root ending in 3. Thus, ∛389017 = 73.

Algebraic Cube Identities

The identities (a + b)³ and (a − b)³ expand a cube without multiplying three lengthy factors separately.

Use (a + b)³ = a³ + 3a²b + 3ab² + b³ and (a − b)³ = a³ − 3a²b + 3ab² − b³. These identities also help recognise numbers close to familiar cubes.

Example: 103³ = (100 + 3)³ = 100³ + 3(100²)(3) + 3(100)(3²) + 3³ = 1,030,301.

Cubes of Negative Numbers

The cube of a negative number is negative, and the cube root of a negative number is also negative.

For every real number a, (−a)³ = −a³ and ∛(−a) = −∛a. The sign remains negative because a negative factor occurs three times.

Example: (−6)³ = −216, so ∛(−216) = −6.

Cube and Cube Root Video Lessons

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Prime Numbers: Test and Find Them Fast

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Practice Cube and Cube Root Questions

Practise published questions related to this topic.

1The LCM of the numbers 1.3, 7.6 and 0.13 is:→ 2Least Common Multiple of A and B is 40 and C = 18. What is the Least Common Multiple of A, B and C?→ 3If the ratio of two numbers is 3 : 8, and their HCF is 6, then their LCM is:→ 4The difference between the two numbers is 1365. On dividing the larger by the smaller, the quotient is 6, and the remainder is 15. Find the smaller number.→ 5If √506.25 = 22.5 , then what is the value of √0.00050625 ?→ 6Three tankers contain 1134 litres, 1215 litres and 513 litres of milk, respectively. Find the capacity of a container (in litres) which can empty the milk of the 3 tankers an exact number of times.→ 7Find the least number that must be added to 10724 to make it a perfect square. Also, find the square root of the resulting perfect square.→ 8What is the smallest 4-digit number that is divisible by 36?→ 9Find the HCF of 44, 110 and 198.→ 10What is the number of ordered pairs (x, y) where x, y are positive integers such that the eight digit number 74x8935y is divisible by 11?→

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Quick Revision Notes

Cube and Cube Root Revision Points

Use these rules and formulas for quick revision.

  • n³ = n × n × n and ∛(n³) = n for every real number n.
  • A perfect cube has prime-factor exponents that are all multiples of 3.
  • The cube of an even number is even, and the cube of an odd number is odd.
  • The unit-digit mapping for a cube root is 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2 and 9→9.
  • In the digit-grouping method, separate digits into groups of three from the right.
  • (a + b)³ = a³ + 3a²b + 3ab² + b³.
  • (a − b)³ = a³ − 3a²b + 3ab² − b³.
  • ∛(−N) = −∛N for N ≥ 0.

Cube and Cube Root FAQs

What is the formula for the cube of a number?

The cube of n is n³ = n × n × n. For example, 4³ = 4 × 4 × 4 = 64.

How do you check whether a number is a perfect cube?

Prime-factorise the number. It is a perfect cube if the exponent of every prime factor is divisible by 3. For example, 1000 = 2³ × 5³ = 10³.

What is the cube root of 13824?

13824 = 2⁹ × 3³ = (2³ × 3)³, so ∛13824 = 2³ × 3 = 24.

How is the cube root of 250047 found by the grouping method?

Write 250047 as 250|047. Since 6³ = 216 and 7³ = 343, the first digit is 6. The last digit 7 gives a root ending in 3, so ∛250047 = 63.

What is the cube root of a negative number?

The cube root remains negative: ∛(−N) = −∛N. Therefore, ∛(−343) = −7.

What is the difference between a cube and a cube root?

A cube is the result of multiplying a number three times, such as 5³ = 125. A cube root reverses this operation, so ∛125 = 5.

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