In competitions such as the JMCWC and MSO World Championship, some questions require calculation of cube roots whose answer is an integer (a whole number).
For example, the cube root of 8000 is 20, because 20³ = 8000.
For example, the cube root of 8765 is 20.6181949…, but that is not an integer. So it doesn’t make sense to find the “exact cube root” of 8765.
As we know that the answer must be an integer, the question is easier, because we can look at both ends of the question separately (the left end and the right end) to easily find different information.
Cubes up to 9³
It is important to know the first integer cubes from memory. Notice that the last digits are all different! And notice how the last digit of the cube is always the same as the last digit of the original number—except for 2 and 8 (they switch), and 3 and 7 (they also switch).
| Number | Cube | Last digit of cube |
| 0 | 0 | 0 |
| 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 |
Cubes of Larger Numbers: Last Digit
Important: you can see immediately what the last digit of any cube is, by looking at the cube of its last digit. For example, what is the last digit of 567³? The last digit of 567 is 7, and 7³ ends in 3 (see table above). Therefore the last digit of 567³ is 3. In fact, 567³ = 182284263.
Optional proof: Any positive integer can be written as a multiple of ten, plus its units digit. For example, 567 = (56 × 10) + 7. In general, it can be written as 10t + u.
Its cube is therefore (10t + u)³ = 1000t³ + 300t²u + 30tu² + u³ = 10(100t³ + 30t²u + 3tu²) + u³
Since 10(100t³ + 30t²u + 3tu²) (without u³) is a multiple of ten, it ends in 0. Therefore the last digit of (10t + u)³ is the same as the last digit of u³.
How to Calculate Exact Cube Roots with 2 Digits in the Answer
The smallest 2-digit number is 10, whose cube 10³ = 1000.
The largest 2-digit number is 99, whose cube is 99³ = 970299.
Therefore, when you find the exact cube root of a 4-, 5- or 6-digit number, the answer is a 2-digit number.
Example: find the exact cube root of 79507
Step 1: find the units digit of the number. In 79507, it’s 7.
Use the table above to find the units digit of the answer. 7 → 3. The last digit is 3.
Step 2: split the rightmost 3 digits and see what remains: From 79507 we have 79.
Use the table above to find the tens digit by finding the cube beneath this. For 79, this is after 64 = 4³ (but before the next cube, 125).
So the tens digit must be 4. We have the units digit from step 1, so the answer is 43.
How to Calculate Exact Cube Roots with more than 2 Digits in the Answer
When you find the cube root of a number with at least 7 digits, the cube root will have at least 3 digits. You can use the method above to find the first and last digits. But you need to use extra techniques to find the middle digit(s). The most suitable method is to use the approximation here: estimating deep roots.
Example: find the exact cube root of 148035889
Step 1: find the units digit of the number. In 148035889, it’s 9.
Use the table above to find the units digit of the answer. 9 → 9. The last digit is 9.
Step 2: starting from the right, split groups of 3 digits and see what remains: From 148035889 we have 148.
Use the table above to find the first digit by finding the cube beneath this. For 148, this is after 125 = 5³ (but before the next cube, 216).
So the first digit must be 5. We have the units digit from step 1, so the answer is 5_9, where the middle digit is still unknown.
Using the method in the link above (not explained here), we estimate (148 – 125) ÷ (3 × 5²) = 0.306666…, so the answer is approximately 530.6666…, and in fact this will be a slight overestimate. In fact, such precision is unnecessary: we could have estimated 0.30 or 0.31 as the approximation instead, and concluded that the answer is close to 530 or 531.
However, we know that the true answer ends in 9. Therefore the true answer is surely 529,
Advanced Techniques when the Task has Too Many Digits
If you need to find the exact cube root of a 10-digit number (or even some 7-digit numbers close to 1000000), this method can be imprecise. Here are some advanced techniques that can help:
- Study a table of the 2-digit endings of cube numbers. This helps you find the last 2 digits (not only the last digit). Maybe one day I’ll add some information about this here 🙂
- Check the remainder when the cube is divided by 3. It should be the same as the remainder when your answer is divided by 3. If you are unsure between 2–3 options for the penultimate digit, this is an easy way to check.
- Memorize some of the 2-digit cubes. This helps you find the two first digits, and allows higher accuracy when finding the next two digits.
- If solving a very large exact cube root, you can use this very difficult mental cube roots algorithm to get as much accuracy as you require.
- There are many other more advanced techniques, such as working modulo various integers. However, start with the techniques above.
Top performers
The fastest people in the world can calculate exact cube roots of 12-digit numbers in a few seconds.
To get in touch with me (Daniel Timms) about mental calculation training, coaching, or anything on this site, you can contact me here.