How to find the cube-root of a number ?



This is an interesting find. I found this shortcut in the most sought after book on Mathematical Shortcuts “Secret of Mental Math” by Arthur Benjamin. Go check it out. It is amazing. We have already seen, how to obtain the cube root of a number whose cube-root is a 2 digit number.

What if the cube-root is a 3-digit number ? How do we obtain the cube root in such cases ?

Before I explain the shortcut, it is very important that you know the concept of digital root or digital sum. If you donot know, please go through it here.

Let me explain the shortcut, by taking a perfect cube having a cube-root which is 3 digit number. Let us assume we need to obtain the cube root of the perfect cube 5088448. Please note that, this shortcut works only for perfect cubes.

Step 1 : First check if the given number is a perfect cube. Since we already know, that 5088448 is a perfect cube, we donot have to check it and we can skip this step. If you do not know how to check if the given number is a perfect cube, please refer here.

Step 2 : Starting from right side, group the numbers into a group of 3 digits. So, we have :

cube-root-1

Step 3 : Since, we have 3 groups, the cube-root will have 3 digits __ __ __ . Now, consider the last group, the number ends in digit 8. So, the cube-root ends in digit 2. So, the cube-root is __ __ 2.

Step 4 : Consider the first group 5. It lies between the perfect cubes 1 and 8. Now, choose the lowest cube-root which is 1 in this case. So, the cube-root is 1 __ 2.

Step 5 : Now, this step is very important. Obtain the digital root of 5088448. The digital root is 8+8+4+8 = 28 = 2+8 = 1.

cube-root-2

Now, from the table, if the digital root of the cube of the number is 1, the digital root of the cube-root of the number could be 1, 4 or 7.

Digital root of thePerfect Cube Digital root ofthe cube root
1 1, 4, 7
2 2, 5, 8
9(0) 3, 6, 9


Now, we need to place that digit in 1 __ 2 so that digital root of this number is 1, 4 or 7. So, the numbers which satisfy this condition is either 1, 4 and 7. So, the cube root is either 112, 142 or 172. Just by inspection we can rule out 112, since 1123 will be closer to 1000000. Now consider 142. Taking a rough estimate, 1423 would 19600 x 14 which close to 2800000. So, the only feasible answer is 172.

So, we can conclude, 1723 = 5088448.

I hope you find this shortcut useful. It might look lengthy at first, but trust me, if you practice this technique, you can easily obtain the answer with 10 seconds.

Let us take another example, so that the shortcut is very clear.

Can you obtain the cube root of the perfect cube 78953589

Step 1 : Let us skip this step as we already know that 78953589 is a perfect cube.

Step 2 : Separate the numbers into a group of 3 digits. Since we have 3 groups, the cube-root will have 3 digits __ __ __.

cube-root-3


Step 3 :
The unit digit of the cube ends in 9. So the unit digit in the cube root also ends in digit 9. So, the cube root is __ __ 9.

Step 4 : Consider the first group 78. This lies between the perfect cubes 64(43) and 125(53). Chose the smallest cube-root 4 which forms the 100’s digit. So, the cube root is 4 __ 9.

Step 5 : Now take the digital root of 78953589. Using Casting 9’s technique, we see that the digital root is 0.

cube-root-5


Now from the table, we see that the digital root of the cube-root 4 __ 9 should either 3, 6 or 9. The numbers which satisfy this condition is either 2, 5 or 8. So, the cube root is either 429, 459 or 489. Just by inspection we can rule out 482, since 4893 will be closer to 125000000. Now consider 4593. Taking a rough estimate, 4523 would be 202500 x 45 which close to 9000000. So, the only reasonable solution is 429.


Using this shortcut, can you obtain the cube-root of the following perfect cubes:

  • 134217728
  • 392223168
  • 3048625
  • 686128968


I hope you find this shortcut useful. Please contact me if certain point is not clear.




Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

9 Comments

  1. Hi,
    Thank you for sharing the short cut.
    However, this method is for finding whose cube root is of 3 digit. What about a number like 65,353,429,952 (whose cube roots are 4 or 5 digits numbers)? I know that last digit will be 8 and first digit will be 4 but now it will make a 4-digit number 4 _ _ 8. Is there any method to find the 2 middle digits?

    • I guess you understood it wrong.

      When you get the digital root of 0, the cube root of the number can have the digital root of 3,6,9. This you understood right ?

      Now, the cube root of 78953589 will have a digital root of either 3,6 or 9. Now we have the cube root 4__9. We need to still find the middle digit. Considering the other digits in the cube root(4 & 9), we need to place that digit in the 4 __ 9 so that we get the digital root of 3, 6 or 9 from the table.

      This is only possible if we put 2,5 or 8

      Just check,

      Digital root(429) = 6
      Digital root(459) = 0
      Digital root(489) = 3

      I hope it is clear.

      ~Kiran

  2. Hi Kiran,

    Thumbs up… keep it up…

    One confusion- in the 1st eg we took the corresponding value of the dig sum of a perfect cube n cube root from the d table i.e. 1 4 7..

    Then in the second case y did we use 2 5 8 as options instead of 3 6 9… the option tried should be 439 469 and 499…

    not able to get this.. pls clarify..
    Thanks
    Reetu

  3. Hi Kiran,

    Really nice work .. Keep posting…

    I have one doubt ..
    In 1st case we too corresponding values (of dig sum cubes n cube roots )from table i.e. 1 4 and 7.. n tried options 112 142 172… (5088448)

    Then in second case why did we not try options 439 469 or 499 b(as in table).. wat made you try 2 5 8… (78953589)
    pls clarify

    Thanks

  4. Hi, sir

    how to find a cube root of 3 digit numbers quickly ?

    for ex-

    1.(233)^3
    2.(256)^3…

    • Please let me know where you came across this question. I haven’t seen any such question in any competitive examination. So I have not written about it.

      Its not that difficult though, but If I come cross any question, I will definitely write about it.

  5. This is a great tip particularly to those new to
    the blogosphere. Brief but very precise information… Thank
    you for sharing this one. A must read article!

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