There are many methods to mentally calculate cube roots (for numbers that are not an exact cube). In this article we explain a quick method that works on any number, and is similar to the method described for mental calculation of square roots.
Introduction:
This method is easier to understand if we first look at the answer squared. As an example, we will find the cube root of 397758. We can write this as:
3√397758 = ab.cdef… = 10*a + 1*b + 0.1*c + 0.01*d + 0.001*e + 0.0001*f + …
Therefore, by forming the cube of the left-hand and right-hand sides:
397758 =
1000 * a³
+ 100 * 3a²b
+ 10 * [3a²c + 3ab²]
+ 1 * [3a²d + 6abc + b³]
+ 0.1 * [3a²e + 6abd + 3ac² + 3b²c]
+ 0.01 * [et cetera]
Notice that the numbers become generally smaller and less significant as we read from the top to the bottom, because of the factors 1000, 100, 10, etc. Therefore we can start by calculating a, subtracting the top line and using the second line to calculate b, subtracting the second line and using the third line to calculate c, etc.
Method:
Step 1:
Calculate the first digit of the answer. Here, it is 7, because:
70³ = 343000 ≤ 397758 < 512000 = 80³
Step 2:
The quantity 3a² appears a lot in the explanation above, so we will calculate this now to help us later:
3a² = 3*7*7 = 147
Step 3:
We have used the first line of the explanation (1000 * a³) to calculate the first digit a, so we can subtract this first line. The first line only uses multiples of 1000, so we can simplify the mental work by using only the largest digits:
397 – 343 = 54
We will next use the second line of the algorithm, which uses multiples of 100, so now we do need to consider the next (fourth) digit of the number 397758. From the explanation, we now have:
547 = 3a²b + [all the other lines, but these are small numbers that we will ignore until later]
To calculate b, we need to divide by 3a², and we know that in this example, 3a² = 147.
547 / 147 = 3 remainder 106
So b = 3, and the answer so far is 73.
Step 4:
We have finished with one line and now move onto the next. Bring the next digit from the original number 397758 onto the remainder:
106 –> 1065
Then the next (third) line of the explanation gives us:
1065 = 3a²c + 3ab² + [all the other lines]
We know a=7 and b=3, so we can subtract the term 3ab² = 3 * 7 * 3 * 3 = 189:
1065 – 189 = 3a² * c + [all the other lines]
876 = 147 * c + [all the other lines]
876 / 147 = 5 (remainder 141)
So c = 5, and the answer so far is 73.5…
Step 5:
Repeat step 4 for every new digit you wish to calculate. In summary, for every new digit:
- multiply the remainder by 10, and add the next digit from the original question
- subtract all terms from the next line of the explanation, except the term with the new unknown letter
- divide the result by 3a². The answer is the next digit of the answer, and the remainder will be used to calculate the next digit.
Sometimes it is necessary to choose a smaller number for the answer so that the (extra-large) remainder will be large enough for subtracting the terms in the following step. See immediately below for an example of this.
To illustrate, here is a continuation of the current example:
- 141 * 10 + 8 = 1418
- 6abc = 6*7*3*5 = 630; 1418 – 630 = 788; b³ = 27; 788 – 27 = 761
- 761 / 147 = 4 (remainder 173)
Note that although 761 / 147 = 5 (remainder 26), this would result in 260 – 630 – 525 – 135 < 0 while calculating the next digit! Therefore we take a smaller dividend (4) to obtain a large remainder. Unfortunately, this happens very frequently during this algorithm.
Back to the correct answer:
- 173 * 10 + 0 = 1730 (+0, because the original question was 397758.0…)
- 6abd = 6*7*3*4 = 504; 1730 – 504 = 1226; 3ac² = 3*7*5*5 = 525; 1226 – 525 = 701; 3b²c = 3*3*3*5 = 135; 701 – 135 = 566
- 566 / 147 = 3 (remainder 125)
This gives us an answer of:
3√397758 = 73.543 (actual answer: 73.542712…)
Summary:
This algorithm is significantly more cumbersome than the equivalent for square roots, and requires some intuition to choose the size needed for the remainders.
Alternative methods—which generalize to fourth-, fifth-, and deeper roots, include:
- a quicker estimation method that provides about 3 digits of accuracy for cube roots.
- a more advanced method involving logarithms.
For practice with cube roots and deeper roots, you can use the Pegasus training software listed on this website.
Inexact cube roots of 6-digit integers was the challenge task in the 2012 Mental Calculation World Cup. Results and challenge questions are here.
If you have any questions or would like specialist coaching from myself, you’re welcome to contact me here.