The cube root of a number is the value that, multiplied by itself three times, gives that number. Because 4 × 4 × 4 = 64, the cube root of 64 is 4, written ∛64 = 4. Cube roots undo cubing in the same way square roots undo squaring. This guide covers four ways to find one: spotting perfect cubes, a last-digit trick that cracks five- and six-digit cubes in seconds, prime factorisation, and estimating roots that are not whole numbers. Keep the cube numbers 1 to 20 chart open alongside; most of the shortcuts lean on it.
Every cube root method below starts here, so these ten are worth learning by heart. Two things make them easy: the cubes of 10, 20, 30 and so on are just the small cube followed by three zeros (20³ = 8,000, 30³ = 27,000), and every last digit from 0 to 9 appears exactly once in the third column. That second fact is the key to the trick in the next section.
| n | n³ | Last digit of n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 8 | 8 |
| 3 | 27 | 7 |
| 4 | 64 | 4 |
| 5 | 125 | 5 |
| 6 | 216 | 6 |
| 7 | 343 | 3 |
| 8 | 512 | 2 |
| 9 | 729 | 9 |
| 10 | 1,000 | 0 |
If you know the number is a perfect cube (a whole number cubed), you can find its cube root in two quick steps.
Units digit of the answer: look at the last digit of the number and match it in the table. 1, 4, 5, 6, 9 and 0 stay the same; 2 and 8 swap; 3 and 7 swap.
Tens digit of the answer: cross off the last three digits. Find the largest cube from the table that is not bigger than what is left. Its root is the tens digit.
Check one: 34 × 34 = 1,156, and 1,156 × 34 = 39,304. The trick only works for perfect cubes up to 999,999 (answers up to 99). If you are not told the number is a perfect cube, check it with the method below or by cubing your answer.
| Number | Last digit → units | Left after removing 3 digits → tens | Cube root |
|---|---|---|---|
| 39,304 | 4 → 4 | 39: 27 ≤ 39 < 64 → 3 | 34 |
| 74,088 | 8 → 2 | 74: 64 ≤ 74 < 125 → 4 | 42 |
| 103,823 | 3 → 7 | 103: 64 ≤ 103 < 125 → 4 | 47 |
| 970,299 | 9 → 9 | 970: 729 ≤ 970 < 1,000 → 9 | 99 |
Prime factorisation works on any perfect cube, however large, and it proves the answer rather than guessing it. Break the number into primes, group the identical primes in threes, and take one from each group.
Example: the cube root of 216,000
216,000 = 216 × 1,000.
216 = 2 × 2 × 2 × 3 × 3 × 3 and 1,000 = 2 × 2 × 2 × 5 × 5 × 5.
Together: 2⁶ × 3³ × 5³, which groups as (2 × 2 × 2)(2 × 2 × 2)(3 × 3 × 3)(5 × 5 × 5).
Take one factor from each group: 2 × 2 × 3 × 5 = 60. Check: 60³ = 216,000.
Example: the cube root of 39,304
Halve repeatedly: 39,304 → 19,652 → 9,826 → 4,913, so 39,304 = 2³ × 4,913.
4,913 is not divisible by 3, 5, 7, 11 or 13, but 4,913 ÷ 17 = 289 and 289 = 17 × 17, so 4,913 = 17³.
39,304 = 2³ × 17³, so the cube root is 2 × 17 = 34, matching the last-digit trick.
If the primes do not fall into complete groups of three, the number is not a perfect cube and its cube root is not a whole number. The same grouping idea works with fractional exponents, since ∛a is a^(1/3); see the exponent rules for how those behave.
Most numbers are not perfect cubes. To find ∛50 by hand, first trap it between two cubes you know: 3³ = 27 and 4³ = 64, so ∛50 is between 3 and 4.
First guess (interpolate): 50 is 23 of the 37 steps from 27 to 64, so try 3 + 23 ÷ 37 ≈ 3.62. Because cubes curve upwards, this guess always comes out a little low.
Improve it (Newton's method): new guess = (2 × old + N ÷ old²) ÷ 3. Start from 3.7: 3.7² = 13.69, 50 ÷ 13.69 = 3.652, (7.4 + 3.652) ÷ 3 = 3.684. Check: 3.684³ ≈ 49.999. One round is usually enough for two decimal places.
Estimating like this is good mental-maths practice; our guide to estimating and rounding has more of the same tricks.
| Number | Between | Interpolated guess | After one Newton step | Actual (3 d.p.) |
|---|---|---|---|---|
| 10 | 2 and 3 | 2.11 | 2.155 (from 2.2) | 2.154 |
| 50 | 3 and 4 | 3.62 | 3.684 (from 3.7) | 3.684 |
Negative numbers have real cube roots. (−3)³ = −27, so ∛(−27) = −3. This is different from square roots, where a negative number has no real root.
Fractions: take the root of the top and bottom separately. ∛(27/125) = ∛27 ÷ ∛125 = 3/5.
Decimals: move the decimal point in steps of three. ∛0.008 = 0.2 because 0.2³ = 0.008, and ∛0.000027 = 0.03.
Zero: ∛0 = 0.
Always put brackets round 1/3. Typing 64^1/3 makes most calculators work out 64¹ ÷ 3 = 21.33 instead of 4. For powers in the other direction, use the exponent calculator.
| Tool | How |
|---|---|
| Scientific calculator | The ∛ key, often a second function above x³; or type the number, then xʸ (or ^) and (1 ÷ 3) |
| TI-84 | Press MATH, choose 4: ∛(, type the number and close the bracket |
| Excel or Google Sheets | =A1^(1/3) or =POWER(A1,1/3); for a negative value use =SIGN(A1)*ABS(A1)^(1/3) |
| Typing the symbol | Copy ∛ (Unicode U+221B), or write cbrt(x) or x^(1/3) |
For quick-fire root practice, play Square Roots or Speed Math, and use the square root calculator for the two-dimensional version of the same idea.
| Find | Answer |
|---|---|
| ∛2,197 (last-digit trick) | 13 |
| ∛21,952 (last-digit trick) | 28 |
| ∛8,000 (cubes of tens) | 20 |
| ∛(−125) | −5 |
| ∛(8/27) | 2/3 |
| ∛20, to one decimal place (estimate) | 2.7 |
For a perfect cube, use the last-digit trick or prime factorisation. For any other number, trap it between two known cubes, make a first guess, then improve it once with (2 × guess + N ÷ guess²) ÷ 3.
60. 216,000 = 216 × 1,000, the cube root of 216 is 6 and the cube root of 1,000 is 10, so the answer is 6 × 10 = 60.
39,304 = 2³ × 17³, so its cube root is 2 × 17 = 34.
Yes. The cube root of a negative number is negative: ∛(−64) = −4, because (−4)³ = −64.
3, because 3 × 3 × 3 = 27.
Learn the first ten cubes and the last-digit pattern, and most perfect cubes up to a million fall in seconds. Use prime factorisation when you need proof, estimation plus one Newton step for everything else, and brackets around 1/3 on a calculator. The full list of cubes to 20³ is on the cube numbers chart.