Cubes of Numbers
The cube of a number is obtained by multiplying the number by itself three times. If a number is n, then its cube is n × n × n = n³.
For example, the cube of 3 is 3 × 3 × 3 = 27. We write this as 3³ = 27. The small 3 written above and to the right is called the exponent, and it tells us to multiply the base by itself 3 times.
A number that can be expressed as the cube of a whole number is called a perfect cube. For example, 8 = 2³, 27 = 3³, 64 = 4³ are all perfect cubes.
The concept of cubes has a strong geometric connection. Just as the square of a number gives the area of a square with that side length, the cube of a number gives the volume of a cube with that side length. A cube-shaped box with side 5 cm has volume 5³ = 125 cm³.
In this topic, you will learn how to find cubes of numbers (including negative numbers and fractions), check whether a number is a perfect cube using prime factorisation, recognise patterns in unit digits of cubes, express cubes as sums of consecutive odd numbers, and find the smallest multiplier or divisor to make a number a perfect cube.
What is Cubes of Numbers?
Definition: The cube of a number n is the product obtained by multiplying n by itself three times.
n³ = n × n × n
Definition: A natural number is called a perfect cube if it can be expressed as the cube of some natural number.
Where:
- n = the base number
- 3 = the exponent (indicates cube)
- n³ = read as "n cubed" or "n raised to the power 3"
Perfect cubes of first 15 natural numbers:
- 1³ = 1
- 2³ = 8
- 3³ = 27
- 4³ = 64
- 5³ = 125
- 6³ = 216
- 7³ = 343
- 8³ = 512
- 9³ = 729
- 10³ = 1,000
- 11³ = 1,331
- 12³ = 1,728
- 13³ = 2,197
- 14³ = 2,744
- 15³ = 3,375
Methods
Method 1: Direct multiplication
- Multiply the number by itself: n × n = n²
- Multiply the result by n again: n² × n = n³
Method 2: Using prime factorisation to check if a number is a perfect cube
- Find the prime factorisation of the number.
- Group the prime factors into triplets (groups of three identical factors).
- If ALL prime factors form complete triplets, the number is a perfect cube.
- If any prime factor is left without a complete triplet, it is NOT a perfect cube.
Example: Is 216 a perfect cube?
- 216 = 2 × 2 × 2 × 3 × 3 × 3
- 216 = 2³ × 3³ = (2 × 3)³ = 6³
- All factors form triplets. So 216 is a perfect cube.
Example: Is 72 a perfect cube?
- 72 = 2 × 2 × 2 × 3 × 3
- 72 = 2³ × 3²
- Factor 3 appears only twice (not a triplet). So 72 is NOT a perfect cube.
Solved Examples
Example 1: Example 1: Finding a cube
Problem: Find the cube of 7.
Solution:
Using n³ = n × n × n:
- 7³ = 7 × 7 × 7
- = 49 × 7
- = 343
Answer: 7³ = 343.
Example 2: Example 2: Cube of a negative number
Problem: Find the cube of (−5).
Solution:
Using n³ = n × n × n:
- (−5)³ = (−5) × (−5) × (−5)
- = 25 × (−5)
- = −125
Answer: (−5)³ = −125.
Example 3: Example 3: Is it a perfect cube?
Problem: Check if 512 is a perfect cube.
Solution:
Prime factorisation:
- 512 = 2 × 256 = 2 × 2 × 128 = 2 × 2 × 2 × 64
- = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
- = 2⁹ = (2³)³ = 8³
All prime factors form complete triplets.
Answer: Yes, 512 is a perfect cube. 512 = 8³.
Example 4: Example 4: Not a perfect cube
Problem: Check if 100 is a perfect cube.
Solution:
Prime factorisation:
- 100 = 2 × 2 × 5 × 5
- = 2² × 5²
Neither 2 nor 5 appears in a group of three.
Answer: No, 100 is not a perfect cube.
Example 5: Example 5: Finding the smallest multiplier
Problem: Find the smallest number by which 108 must be multiplied to make it a perfect cube.
Solution:
Prime factorisation:
- 108 = 2 × 2 × 3 × 3 × 3
- = 2² × 3³
The factor 2 appears only twice. It needs one more 2 to form a triplet.
- 108 × 2 = 216 = 2³ × 3³ = 6³
Answer: The smallest number is 2.
Example 6: Example 6: Finding the smallest divisor
Problem: Find the smallest number by which 1536 must be divided to make it a perfect cube.
Solution:
Prime factorisation:
- 1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3
- = 2⁹ × 3¹
2 forms complete triplets (2⁹ = (2³)³). But 3 appears only once.
- Divide by 3: 1536 ÷ 3 = 512 = 2⁹ = 8³
Answer: The smallest number is 3.
Example 7: Example 7: Cube of a fraction
Problem: Find the cube of 2/3.
Solution:
Using (a/b)³ = a³/b³:
- (2/3)³ = 2³ / 3³
- = 8 / 27
Answer: (2/3)³ = 8/27.
Example 8: Example 8: Cube of a decimal
Problem: Find the cube of 0.4.
Solution:
Converting to fraction:
- 0.4 = 4/10 = 2/5
- (2/5)³ = 2³/5³ = 8/125
- = 0.064
Answer: (0.4)³ = 0.064.
Example 9: Example 9: Sum of consecutive odd numbers
Problem: Express 5³ as the sum of consecutive odd numbers.
Solution:
The cube of n is the sum of n consecutive odd numbers.
For n = 5: Start from the (n(n−1) + 1)th odd number = (5 × 4 + 1)th = 21st odd number = 41.
- 5³ = 41 + 43 + 45 + 47 + 49
- = 225 − 100 = 125 ✓
Verification: 41 + 43 + 45 + 47 + 49 = 225... Actually 41+43=84, +45=129, +47=176... let us recheck.
- 41 + 43 = 84
- 84 + 45 = 129
- Wait — that gives 129, not 125.
Correct starting point: For n = 5, use: 21 + 23 + 25 + 27 + 29 = 125.
- 21 + 23 = 44
- 44 + 25 = 69
- 69 + 27 = 96
- 96 + 29 = 125 ✓
Answer: 5³ = 21 + 23 + 25 + 27 + 29 = 125.
Example 10: Example 10: Unit digit of a cube
Problem: What is the unit digit of 23³?
Solution:
The unit digit depends only on the unit digit of the base.
- Unit digit of 23 is 3.
- From the pattern: a number ending in 3 has a cube ending in 7.
- (Check: 3³ = 27, unit digit = 7 ✓)
Answer: The unit digit of 23³ is 7.
Real-World Applications
Real-world applications of cubes:
- Volume calculation: The volume of a cube-shaped box = side³. A cube with side 5 cm has volume 5³ = 125 cm³.
- Capacity measurement: Finding the volume of cuboidal tanks, containers, and rooms uses the concept of cubing dimensions.
- Physics: The relationship between mass, volume, and density uses cubes when dimensions are involved.
- Computer science: Memory and storage sizes often involve powers of 2, including cubes (e.g., 2¹⁰ = 1024).
- Architecture: Estimating material for cubic structures — bricks, concrete blocks, etc.
- Number theory: Perfect cubes appear in many mathematical puzzles and patterns.
- Chemistry: Crystal structures are often cubic, and understanding unit cells involves cubing edge lengths.
Key Points to Remember
- The cube of n is n³ = n × n × n.
- A perfect cube is a number that can be written as n³ for some integer n.
- Cube of an even number is even; cube of an odd number is odd.
- Cube of a negative number is negative.
- To check if a number is a perfect cube, do prime factorisation and check that every prime forms a triplet.
- The unit digit of n³ depends only on the unit digit of n.
- Pairs (2, 8) and (3, 7) swap unit digits when cubed.
- Numbers ending in 0, 1, 4, 5, 6, 9 retain their unit digit when cubed.
- n³ can be expressed as the sum of n consecutive odd numbers.
- (a/b)³ = a³/b³ — the cube of a fraction = cube of numerator / cube of denominator.
Practice Problems
- Find the cube of 13.
- Is 729 a perfect cube? If yes, find the number whose cube it is.
- Find the cube of (−8).
- Find the smallest number by which 392 must be multiplied to make it a perfect cube.
- What is the unit digit of 37³?
- Find the cube of 3/4.
- Is 500 a perfect cube? If not, find the smallest number by which it should be divided to make it a perfect cube.
- Express 4³ as the sum of consecutive odd numbers.
Frequently Asked Questions
Q1. What is the cube of a number?
The cube of a number n is n × n × n, written as n³. It represents the volume of a cube with side length n.
Q2. What is a perfect cube?
A perfect cube is a number that equals n³ for some whole number n. Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Q3. How do you check if a number is a perfect cube?
Find the prime factorisation of the number. If every prime factor appears in groups of three (triplets), it is a perfect cube. If any factor is left over, it is not.
Q4. Is the cube of a negative number positive or negative?
The cube of a negative number is always negative. For example, (−4)³ = −64. This is because negative × negative = positive, then positive × negative = negative.
Q5. What is the difference between square and cube?
A square is n × n (n²), and a cube is n × n × n (n³). Square gives area; cube gives volume. Example: 5² = 25, 5³ = 125.
Q6. What is the cube of 0?
0³ = 0 × 0 × 0 = 0. The cube of zero is zero.
Q7. How do you find the unit digit of a cube?
The unit digit of n³ depends only on the unit digit of n. Use the pattern: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9.
Q8. Can a perfect cube end in any digit?
Yes. Unlike perfect squares (which cannot end in 2, 3, 7, 8), perfect cubes can end in any digit from 0 to 9.
Q9. What are the first 10 perfect cubes?
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 — these are the cubes of 1 through 10.
Q10. What is the cube of a fraction?
(a/b)³ = a³/b³. Cube the numerator and denominator separately. Example: (2/5)³ = 8/125.
