Maths
What are cube numbers and cube roots, and how do you find a cube root?
Quick answer
A cube number is a number multiplied by itself three times, like 4³ = 4 × 4 × 4 = 64. The cube root undoes it, so ∛64 = 4.
In one sentence
A cube number is what you get when you multiply a whole number by itself three times, and the cube root takes you back to the number you started with.
Explained
Build a cube out of small building blocks. If it is 2 blocks long, 2 wide and 2 tall, it uses 2 × 2 × 2 = 8 blocks. A 3-block cube uses 3 × 3 × 3 = 27 blocks. That is where the name comes from: cube numbers are the numbers of blocks that make perfect cubes, just as square numbers make perfect squares.
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We write a cube with a small 3, called an index or power: n³ = n × n × n, read "n cubed". So 5³ = 5 × 5 × 5 = 125. It is also the volume of a cube with side n: a cube with 5 cm edges holds 125 cm³.
The cube root goes the other way. It answers the question "which number, cubed, gives this?" Its symbol is a root sign with a small 3: ∛.
If n³ = x, then ∛x = n. For example, 4³ = 64, so ∛64 = 4.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1,000 |
| n | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 1,331 | 1,728 | 2,197 | 2,744 | 3,375 | 4,096 | 4,913 | 5,832 | 6,859 | 8,000 |
A few things make cubes different from squares:
- Negative numbers. Three minus signs multiply to a minus, so (−3)³ = −27. That means a negative number has a cube root: ∛(−27) = −3. (A negative number has no square root among ordinary numbers.)
- One answer only. 25 has two square roots, 5 and −5, but 125 has only one cube root, 5.
- Fractions and decimals. (1/2)³ = 1/8 and 0.2³ = 0.008. Cubing a number between 0 and 1 makes it smaller.
Most numbers are not perfect cubes, so their cube roots are not whole numbers. ∛3 ≈ 1.442 and ∛50 ≈ 3.684. You can always estimate: 50 lies between 27 (3³) and 64 (4³), so ∛50 is between 3 and 4. Cubes are a special case of powers, and cube roots are the cousins of square roots.
Examples
- 7³ = 7 × 7 × 7 = 49 × 7 = 343
- (−4)³ = (−4) × (−4) × (−4) = 16 × (−4) = −64
- ∛216 = 6, because 6 × 6 × 6 = 216
- ∛1,000 = 10, and ∛1,000,000 = 100
- (2/3)³ = (2 × 2 × 2) ÷ (3 × 3 × 3) = 8/27
- Two nice patterns: 1 = 1³, 3 + 5 = 8 = 2³, 7 + 9 + 11 = 27 = 3³ (groups of odd numbers), and 1³ + 2³ + 3³ = 36 = (1 + 2 + 3)².
Common mistakes
- Multiplying by 3 instead of cubing. 4³ is 4 × 4 × 4 = 64, not 4 × 3 = 12.
- Stopping after two multiplications. 5 × 5 = 25 is 5², not 5³. Fix: write all three 5s.
- Thinking ∛(−8) has no answer. It is −2, because (−2)³ = −8.
- Cubing a sum piece by piece. (2 + 3)³ = 5³ = 125, but 2³ + 3³ = 8 + 27 = 35. Add first, then cube.
Step by step
Step by step
- Check the last digitEvery digit from 0 to 9 gives a different last digit when cubed: 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0. So the last digit of a perfect cube tells you the last digit of its cube root.
- Cover the last three digitsLook at what is left. For 13,824, that is 13.
- Find the tens digitFind the biggest cube that is not more than what is left: 2³ = 8 ≤ 13 < 27, so the tens digit is 2.
- Put it together13,824 ends in 4, so the root ends in 4. ∛13,824 = 24.
- Check24 × 24 × 24 = 576 × 24 = 13,824.
This trick only works for perfect cubes (up to 6 digits). For other numbers, estimate between two cubes or use a calculator: press the ∛ key (often Shift then √), or type the number to the power of (1 ÷ 3). In a spreadsheet, =POWER(A1,1/3) does the same.
Practice questions
Cube numbers and cube roots
Try each one, then check your answer.
- Work out 3³.
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3 × 3 × 3 = 27 - Work out 6³.
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6 × 6 × 6 = 36 × 6 = 216 - Work out 10³.
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1,000 - Find ∛8.
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2 × 2 × 2 = 8, so 2 - Find ∛343.
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7 × 7 × 7 = 343, so 7 - Find ∛729.
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9 × 9 × 9 = 729, so 9 - Work out (−5)³.
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(−5) × (−5) × (−5) = 25 × (−5) = −125 - Find ∛(−64).
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(−4)³ = −64, so −4 - Work out 0.2³.
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0.2 × 0.2 × 0.2 = 0.008 - Work out (2/3)³.
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8 ÷ 27 = 8/27 - Is 100 a cube number?
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4³ = 64 and 5³ = 125, so no. - Between which two whole numbers is ∛50?
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3³ = 27 and 4³ = 64, so between 3 and 4 (about 3.68). - Find ∛1,728.
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12 × 12 × 12 = 1,728, so 12 - Find ∛4,096.
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16 × 16 × 16 = 4,096, so 16 - Find ∛13,824 without a calculator.
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Ends in 4, so the root ends in 4; 13 is between 2³ and 3³, so 24. - Which is bigger, 2³ + 3³ or (2 + 3)³?
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8 + 27 = 35 and 5³ = 125, so (2 + 3)³ is bigger.
Word problems
Cubes in real life
Try each one, then check your answer.
- A cube-shaped box has a volume of 1,000 cm³. How long is each edge?
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∛1,000 = 10 cm - A big cube is built from small cubes, 3 along each edge. How many small cubes are used?
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3³ = 27 - A cube-shaped water tank has 2 m edges. How many litres does it hold? (1 m³ = 1,000 litres)
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2³ = 8 m³, so 8,000 litres - An ice cube has a volume of 27 cm³. How long is each edge?
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∛27 = 3 cm - A wooden cube with 5 cm edges is painted on the outside, then cut into 1 cm cubes. How many small cubes have no paint at all?
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The unpainted ones form a 3 × 3 × 3 cube inside: 27 - The edge of a cube doubles from 3 cm to 6 cm. How many times bigger is its volume?
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216 ÷ 27 = 8 times (2³ = 8)
Questions people ask
What is a cube number?
A cube number is a whole number multiplied by itself three times. 1, 8, 27, 64 and 125 are the first five cube numbers.
What are the cube numbers from 1 to 10?
1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000. The tables above go on to 20³ = 8,000.
What is the cube root of 64, 125 and 216?
∛64 = 4, ∛125 = 5 and ∛216 = 6. Each one cubed gives the number back.
How do you find a cube root without a calculator?
For a perfect cube, use the last digit to find the units digit of the answer, then cover the last three digits to find the tens digit. For any other number, find the two cubes it lies between and estimate.
How do you do a cube root on a calculator?
Use the ∛ key, which is often Shift and then √. On a calculator without one, use the power key: type the number, then ^ (1 ÷ 3).
Can a cube root be negative?
Yes. The cube root of a negative number is negative: ∛(−125) = −5, because (−5)³ = −125.
Which numbers are both square and cube numbers?
1, 64, 729 and 4,096. They are the sixth powers: 1⁶, 2⁶, 3⁶ and 4⁶. For example, 64 = 8² = 4³.
What is the cube root of 3?
About 1.442. It is not a whole number or a fraction, so it never ends or repeats as a decimal.
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Related topics
Square numbers
A square number is what you get when you multiply a whole number by itself, like 4 × 4 = 16. The first square numbers are 1, 4, 9, 16, 25 and 36.
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Highest common factor and lowest common multiple
The HCF is the largest number that divides exactly into two numbers. The LCM is the smallest number that both divide into. For 12 and 18, the HCF is 6 and the LCM is 36.
Logarithms
A logarithm answers the question "what power?". log₂ 8 = 3 because 2³ = 8. Logs undo exponents, and their rules turn multiplying into adding.