How To Mentally Calculate Cube Roots As 2-Digit Integers?

Wouldn’t it be cool if you were able to quickly calculate cube roots as 2-digit integers (the cube roots of perfect cubes of all 2-digit integers) in your head? If you are a math enthusiast, such a neat little trick would be right up your alley.

Cube roots are typically considered complex calculations. They are normally not suited for mental calculations. Yet, most recently, I stumbled upon a very old algorithm that enables quick mental computations of cube roots.

The best part is that this trick makes you look like a math wizard without actually having to be one. I am yet to come across any person in my life who has pulled this off.

Without any further ado, let’s get started with it.

This essay is supported by Generatebg

How to Compute Cube Roots As 2-Digit Integers

The First Two Steps

The first step that you’ll need to do is the hard part. But it is a one-time effort, and in my opinion, well worth it. You will have to memorize the cubes of the first ten integers starting from 1 up to 10.

The second part of the trick is to note that the ending digits of the cubes are the same as the cubed number for 1,4,5,6,9, and 10. Furthermore, the ending digits of the cubes are exchanged between the pairs 2–8 and 3–7 respectively. Once you have this part nailed down, you are almost there.

The Final Step

Now, let us consider the question from the title image. What is the cube root of 175616? Let us focus on the last digit. It is ‘6’. Based on our previous observation, we know that the cube of 6 ends in 6.

We can use this to our advantage to predict that the cube root of 175616 will end in 6. With that, we have the one’s place of the cube root figured out.

Next, just ignore the last three digits of 175616 (that is, 616) and focus on the first three (that is, 175). Next, ask the following question:

“Which integer from 1 to 10 when cubed is closest to but not greater than 175?”

We know that 5³ = 125 is lower than 175 and 6³ = 216 is greater than 175. So, 5 would be the right answer to our question. So, you just write 5 in front of 6. And there you have it. 56 is your answer!

At this point, if you are the typical math enthusiast, I am sure that you are dying to get to know WHY this works! Worry not. We shall explore the mathematics behind this algorithm next.

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The Mathematics Behind the Cube Roots Algorithm

One’s Place

I will be using algebra to demonstrate the mathematics behind this algorithm.

The first significant observation is that any 2-digit integer could be expressed in the form of the following expression:

Now, let us use the binomial theorem to cube this expression:

When we look at the expansion of the cubed expression, a few interesting properties of this arrangement reveal themselves. Firstly, the y³ part is going to occupy one’s place in the final result. This is the reason why we can confidently deduce the one’s place number by just looking at the last digit of the number for which we are seeking the cube root.

To see this effect more clearly, let us consider the number 56. As per our initial observation, this could be written as follows:

When we try to add the terms using the conventional addition algorithm, it becomes clear that the sum of the first three terms (1000x³ + 300x²y + 30 xy²) needs to necessarily end in a zero. Therefore, y³ is the only term that contributes to the one’s place of the cubed number.

From the ending digit of the original number (175616), we have thus far deduced ‘y’. Our task now is to deduce ‘x’.

Ten’s place

To this end, let us analyse the first three terms: ‘1000x³ + 300x²y + 30 xy²’. When we look at the summation again, we see that we could take advantage of the fact that the first term (1000*x³) is the biggest contributor to the sum, and contains ‘x’ exclusively.

When we divide the result of the sum by 1000 (in order to narrow down our focus on just ‘x’), we end up focusing on just the first three digits of the sum (in this case 175).

Because of the contribution of the terms: ‘300x²y + 30xy²’, we know for sure that the number formed by the first three digits of the sum would be necessarily greater than x³. So, we could treat x³ as a lower bound. Based on this information alone, we can start guessing ‘x’. This is because cubes of all integers from 1 to 9 (that we have memorized at this point) do not exceed three digits.

But still, we need to ensure that we consider the upper bound as well. To do this, let us consider ‘(x+1)³’ as the upper bound. We could now tabularize the terms ‘[1000x³ + 300x²y + 30xy²]/1000’ and ‘(x+1)³’ against each other for all single-digit integer values of ‘x’ and the highest possible value of ‘y’ (which is 9) to see the limits of the upper bound as follows:

As a result, we see clearly that for the highest possible ‘y’ value, in combination with all possible x values, we get a consistent result where the expression ‘[1000x³ + 300x²y + 30xy²]/1000’ is greater than x³ and less than (x+1)³.

This is the reason why we just look at the first three digits of the number (175616) and try to recall the closest number whose cube value is closest to but not greater than the number formed by the first three digits.

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Summary and Application for Fun

The algorithm to mentally calculate cube roots as 2-digit integers can be summarized as follows:

1. Memorise cube values of integers from 1 through 10.

2. Note that the ending digits of the cubes are the same as the cubed number for 1,4,5,6,9, and 10. The ending digits are exchanged between the cube pairs of 2–8 and 3–7 respectively.

3. Look at the last digit of the number for which you seek the cube root. Use the results from the above step to arrive at the one’s place of the cube root.

4. Look at the first three digits of the number for which you seek the cube root. Recall the number whose cube value is closest to but not greater than the number formed by these three digits.

Now that you have come so far, it is time for some fun! You could employ this method for your entertainment at a party as follows:

1. First, gather a nerdy crowd at the party, and choose a random person from the crowd.

2. Then, ask this person to guess any two-digit integer (and not reveal it to you at this stage).

3. Next, ask this person to cube this integer and reveal the final result (only) to you.

4. Employ the algorithm and reveal (to the crowd) the number that the person had originally guessed.

5. Enjoy the reaction of the crowd, and put on a smug face for added entertainment!

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Reference: Center for Bibliographical Studies and Research (San Francisco Call — 1913).

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Further reading that might interest you: How To Mentally Square Any Integer Ending in 5? and How To Quickly Calculate Percentages In The Head?