What is a cube number?
Take small blocks of the same size. Put n blocks along the length, n along the width and n up the height. The big cube has n × n × n blocks. We write this as n³ and say "n cubed".
- 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000
Why the name? These numbers of blocks always make a perfect cube shape. The same idea gives volume: a cube with side a cm has volume a³ cm³.
Try it: stack sugar cubes or dice into a 2 × 2 × 2 cube. Count them. Then try 3 × 3 × 3.
Perfect cubes and their patterns
A perfect cube is a whole number that equals some whole number cubed. 216 is a perfect cube (6³). 100 is not.
Last digits
The cube of a number ending in 1, 4, 5, 6, 9 or 0 ends in the same digit. Ends in 2 → cube ends in 8, and 8 → 2. Ends in 3 → cube ends in 7, and 7 → 3.
Odd and even
The cube of an even number is even. The cube of an odd number is odd.
Negative numbers
(−2)³ = −8, so the cube of a negative number is negative. That is why ∛(−8) = −2 makes sense, unlike square roots.
Finding cube roots
The cube root of m, written ∛m, is the number whose cube is m.
Method 1: prime factorisation
Break the number into prime factors. Make groups of three equal factors. Take one factor from each group and multiply.
3375 = 3 × 3 × 3 × 5 × 5 × 5 = (3 × 5)³, so ∛3375 = 15.
If some factor is left without a full group of three, the number is not a perfect cube. Multiply (or divide) by the missing (or extra) factors to make it one.
Method 2: estimation (perfect cubes up to 6 digits)
Split off the last three digits. The last digit of the cube tells the last digit of the root. The front part tells the tens digit: find the biggest cube that is not more than it.
For 17576: it ends in 6, so the root ends in 6. Front part 17 lies between 2³ = 8 and 3³ = 27, so the tens digit is 2. ∛17576 = 26.
Sums of cubes and the number 1729
Some numbers are the sum of two cubes: 9 = 1³ + 2³, 35 = 2³ + 3³.
1729 can be written this way in two different ways: 1729 = 1³ + 12³ = 9³ + 10³. It is the smallest such number. It is called the Hardy–Ramanujan number or a taxicab number. The story: G. H. Hardy came to visit in a taxi numbered 1729 and called it a dull number. Srinivasa Ramanujan at once said it was very interesting, for this reason.
Another pattern: 1³ + 2³ + 3³ = 36 = (1 + 2 + 3)². Adding the first few cubes always gives a square number.
Key formulas and definitions
- n³ = n × n × n
- Volume of a cube = a³
- ∛(n³) = n
- ∛(a × b) = ∛a × ∛b
- (−n)³ = −n³
Worked examples
1. Find 7³.
7 × 7 = 49, and 49 × 7 = 343. So 7³ = 343.
2. Is 216 a perfect cube? If yes, find its cube root.
216 = 2 × 2 × 2 × 3 × 3 × 3. Groups: (2,2,2) and (3,3,3). All factors are in triplets, so yes. ∛216 = 2 × 3 = 6.
3. Find ∛2744 by prime factors.
2744 = 2 × 2 × 2 × 7 × 7 × 7. One 2 from the first triplet, one 7 from the second. ∛2744 = 2 × 7 = 14.
4. Is 500 a perfect cube? What is the smallest number to multiply it by to make one?
500 = 2 × 2 × 5 × 5 × 5. The 5s form a triplet but 2 appears only twice. Multiply by one more 2: 500 × 2 = 1000 = 10³. Answer: 2.
5. Estimate ∛13824 (it is a perfect cube).
It ends in 4, so the root ends in 4. Front part 13: 2³ = 8 ≤ 13 < 27 = 3³, so the tens digit is 2. ∛13824 = 24. Check: 24³ = 13824.
6. Find ∛(−1331) and ∛(27/125).
11³ = 1331, so ∛(−1331) = −11. ∛27 = 3 and ∛125 = 5, so ∛(27/125) = 3/5.
Common mistakes
- Thinking n³ = 3 × n. 4³ is 4 × 4 × 4 = 64, not 12.
- Grouping prime factors in pairs (that is for square roots). For cube roots, group in threes.
- Saying a negative number has no cube root. ∛(−27) = −3 because (−3)³ = −27.
- Using the last-digit trick on numbers that are not perfect cubes. It only works when you know the number is a perfect cube.