Vedic Math - Cube Roots

Note: Vedic Math Blog has been moved to http://vedicmath.vedantatree.com/. Please bookmark the new address for new and existing blogs.

Today, we are taking a next level of topic i.e. finding the cube root. With normal approach, finding cube root is bit complex. However, using Vedic Math techniques, it becomes interesting and fast too. This amazing technique will help you to find out the cube root of a  4 or 5 or 6 digits number quickly and all using mind power only. Technique specified in this article will work for perfect cubes only, not for other numbers (that we shall discuss in forthcoming articles). Lets start learning.

We know that, cube of a 2-digit number will have at max 6 digits (99³ = 970,299). This implies that if you are given with a 6 digit number, its cube root will have 2 digits. Further, following are the points to remember for speedy calculation of cube roots (of perfect cubes).

1. The lowest cubes (i.e. the cubes of the fist nine natural numbers) are 1, 8, 27, 64, 125, 216, 343, 512 and 729.
2. They all have their own distinct endings; with no possibility of over-lapping (as in the case of squares).
3. The last digit of the cube root of an exact cube is obvious:
• 1³ = 1    > If the last digit of the perfect cube = 1, the last digit of the cube root = 1
• 2³ = 8    > If the last digit of the perfect cube = 8, the last digit of the cube root = 2
• 3³ = 27  > If the last digit of the perfect cube = 7, the last digit of the cube root = 3
• 4³ = 64  > If the last digit of the perfect cube = 4, the last digit of the cube root = 4
• 5³ = 125 > If the last digit of the perfect cube = 5, the last digit of the cube root = 5
• 6³ = 216 > If the last digit of the perfect cube = 6, the last digit of the cube root = 6
• 7³ = 343 > If the last digit of the perfect cube = 3, the last digit of the cube root = 7
• 8³ = 512 > If the last digit of the perfect cube = 2, the last digit of the cube root = 8
• 9³ = 729 > If the last digit of the perfect cube = 9, the last digit of the cube root = 9
4. In other words,
• 1, 4, 5, 6, 9 and 0 repeat themselves as last digit of cube.
• Cube of 2, 3, 7 and 8 have complements from 10 (e.g. 10's complement of 3 is 7 i.e. 3+7=10) as last digit.
5. Also consider, that 
• 8's cube ends with 2 and 2's cube ends with 8 
• 7's cube ends with 3 and 3's cube ends with 7

If we observe the properties of numbers, Mathematics becomes very interesting subject and fun to learn. Following same, let’s now see how we can actually find the cube roots of perfect cubes very fast.

Example 1:  Find Cube Root of 13824

Step 1:
Identify the last three digits and make groups of three digits from right side. That is 13824 can be written as          
   13  ,   824
 
Step 2: 
Take the last group which is 824.  The last digit of 824 is 4.
Remember point 3, If the last digit of the perfect cube = 4, the last digit of the cube root = 4
Hence the right most digit of the cube root  = 4

Step 3:
Take the next group which is 13.
From point 3, we see that 13 lies between 8 and 27 which are cubes of 2 and 3 respectively. So we will take the cube root of the smaller number i.e. 8 which is 2.
So 2 is the tens digit of the answer.

We are done and the answer is '24'

Isn't that easy and fun..