In problems of algebra dealing with factorization, equations of third power and in many problems of geometry related to three dimensional figures, one will often find the need to calculate the cube root of numbers. Calculating the cube root of a number by the traditional method is a slightly cumbersome procedure, but the technique used by Vedic Mathematicians is so fast that one can get the answer in two to three seconds!

The technique for solving cube roots is simply so amazing that the student will be able to correctly predict the cube root of a number just by looking at it and without the need for any intermediate steps.

You might find it difficult to believe, but at the end of this study, you will be calculating cube roots of complicated numbers like 262144, 12167 and 117649 in seconds. Even primary school students who have learnt these techniques from me are able to calculate cube roots in a matter of seconds.
Before we delve deeper in this study, let us clear our concepts relating to cube roots.

WHAT IS CUBE ROOTING?

Let us take the number 3. When we multiply 3 by itself we are said to have squared the number 3. Thus 3 * 3 is 9. When we multiply 4 by itself we are said to have squared 4 and thus 16 is the square of 4.

Similarly the square of 5 is 25 (represented as 5²) The square of 6 is 36 (represented as 6²)

In squaring, we multiply a number by itself, but in cubing we multiply a number by itself and then multiply the answer by the original number once again.

Thus, the cube of 2 is 2 * 2 * 2 and the answer is 8.

(represented as 2³)

The cube of 3 is 3 * 3 * 3 and the answer is 27.

(represented as 3³)

Basically, in squaring we multiply a number by itself and in cubing we multiply a number twice by itself.

Now, since you have understood what cubing is it will be easy to understand what cube rooting is. Cube rooting is the procedure of determining the number which has been twice multiplied by itself to obtain the cube. Calculating the cube root is the reciprocal procedure of calculating a cube.

Thus, if 8 is the cube of 2, then 2 is the cube root of 8.

If 27 is the cube of 3 then 3 is the cube root of 27 and so on.

In this chapter, we will learn how to calculate cube roots. Thus, if you are given the number 8 you will have to arrive at the number 2. If you are given the number 27 you will have to arrive at 3. However, these are very basic examples. We shall be cracking higher-order numbers like 704969, 175616, etc.

At this point, I would like to make a note that the technique provided in this chapter can be used to find the cube roots of perfect cubes only. It cannot be used to find the cube root of imperfect cubes.

METHOD

I have given below a list containing the numbers from 1 to 10 and their cubes. This list will be used for calculating the cube roots of higher-order numbers. With the knowledge of these numbers, we shall be able to solve the cube roots instantly. Hence, I urge the reader to memorize the list given below before proceeding ahead with the chapter.

NUMBER	CUBE
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000

The cube of 1 is 1, the cube of 2 is 8, the cube of 3 is 27 and so on…
Once you have memorized the list I would like to draw your attention to the underlined numbers in the key. You will notice that I have underlined certain numbers in the key. These underlined numbers have a unique relationship among themselves.

In the first row, the underlined numbers are 1 and 1. It establishes a certain relationship that if the last digit of the cube is 1 then the last digit of the cube root is also 1.

In the second row, the underlined numbers are 2 and 8. It establishes a relationship that if the last digit of the cube is 8 then the last digit of the cube root is 2. Thus, in any given cube if the last digit of the number is 8 the last digit of its cube root will always be 2.

In the third row, the underlined numbers are 3 and 7 (out of 27 we are interested in the last digit only and hence we have underlined only 7). We can thus conclude that if the last digit of a cube is 7 the last digit of the cube root is 3.

And like this if we observe the last row where the last digit of 10 is 0 and the last digit of 1000 is also 0. Thus, when a cube ends in 0 the cube root also ends in 0.

On the basis of the above observations, we can form a table as given below:

The Last Digit Of The Cube	The Last Digit Of The Cube Root
1	1
2	8
3	7
4	4
5	5
6	6
7	3
8	2
9	9
0	0

From the above table, we can conclude that all cube roots end with the same number as their corresponding cubes except for pairs of 3 & 7 and 8 & 2 which end with each other.

There is one more thing to be kept in mind before solving cube roots:
Whenever a cube is given to you to calculate its cube root, you must put a slash before the last three digits.

If the cube given to you is 103823 you will represent it as 103 823
If the cube given to you is 39304, you will represent it as 39 304

Regardless of the number of digits in the cube, you will always put a slash before the last three digits.

SOLVING CUBE ROOTS

No.	1	2	3	4	5	6	7	8	9	10
Cube	1	8	27	64	125	216	343	512	729	1000

We will be solving the cube root in 2 parts. First, we shall solve the RHS of the answer and then we shall solve the LHS of the answer. If you wish you can solve the LHS before the RHS. There is no restriction on either method but generally people prefer to solve the RHS first.

As illustrative examples, we shall take four different cubes.

(Q) Find the cube root of 287496.

•We shall represent the number 287496 as 287 496

•Next, we observe that the cube 287496 ends with a 6 and we know that when the cube ends with a 6, the cube root also ends with a 6. Thus our answer at this stage is 6. We have thus got the RHS of our answer.

•To find the LHS of the answer we take the number which lies to the left of the slash. In this case, the number lying to the left of the slash is 287. Now, we need to find two perfect cubes between which the number 287 lies in the number line. From the key, we find that 287 lies between the perfect cubes 216 (the cube of 6) and 343 (the cube of 7).

•Now, out of the numbers obtained above, we take the smaller number and put it on the LHS of the answer. Thus, out of 6 and 7, we take the smaller number 6 and put it beside the answer of 6 already obtained. Our final answer is 66. Thus, 66 is the cube root of 287496.

(Q) Find the cube root of 205379.

•We represent 205379 as 205/379

•The cube ends with a 9, so the cube root also ends with a

9.(The answer at this stage is 9.)

•The part to the left of the slash is 205. It lies between the perfect cubes 125 (the cube of 5) and 216 (the cube of 6)

•Out of 5 and 6, the smaller number is 5 and so we take it as the left part of the answer. The final answer is 59.

(Q) Find the cube root of 681472.

•We represent 681472 as 681/472

•The cube ends with a 2, so the root ends with an 8. The answer at this stage is 8.

•681 lies between 512 (the cube of 8) and 729 (the cube of 9).

•The smaller number is 8 and hence our final answer is 88.

(Q)Find the cube root of 830584.

•The cube ends with a 4 and the root will also end with a 4.

•830 lies between 729 (the cube of 9) and 1000 (the cube of 10).

•Since the smaller number is 9, the final answer is 94.

You will observe that as we proceeded with the examples, we took much less time to solve the cube roots. After some practice you will be able to solve the cube roots by mere observation of the cube and without needing any intermediary steps.

It must be noted that regardless of the number of digits in the cube, the procedure for solving them is the same.

(Q)Find the cube root of 2197.

•The number 2197 will be represented as 2/197
•The cube ends in 7 and so the cube root will end with a 3.

We will put 3 as the RHS of the answer.

•The number 2 lies between 1 (the cube of 1) and 8 (the cube of 2).
•The smaller number is 1 which we will put as the

LHS of the answer. The final answer is 13.

We may thus conclude that there exists only one common procedure for solving all types of perfect cube roots.

In my seminars, the participants often ask the procedure for solving cube roots of numbers having more than six digits. (All the examples that we have solved before had six or fewer digits.)

Well the answer to this question is that the procedure for solving the problem is the same. The only difference in this case is that you will be expanding the number line.

Let us take an example.

We know that the cube of 9 is 729 and the cube of 10 is 1000. Now let us go a step ahead and include the higher numbers. We know that the cube of 11 is 1331 and the cube of 12 is 1728.

Number	9	10	11	12
Cube	729	1000	1331	1728

(Q) Find the cube root of 1157625.

•We put a slash before the last three digits and represent the number as 1157 625.

•The number 1157625 ends with a 5 and so the root also ends with a 5. The answer at this stage is 5.

•We take the number to the left of the slash, which is 1157. In the number line it lies between 1000 (the cube of 10) and 1331 (the cube of 11).

•Out of 10 and 11, we take the smaller number 10 and put it beside the 5 already obtained. Our final answer is 105.

(Q) Find the cube root of 1404928.

•The number will be represented as 1404/928

•The number ends with a 8 and so the cube root will end with a 2.

•1404 lies between 1331 (the cube of 11) and 1728 (the cube of 12). Out of 11 and 12 the smaller number is 11 which we will put beside the 2 already obtained. Hence, the final answer is 112.

The two examples mentioned above were just for explanation purposes. Under normal circumstances, you will be asked to deal with cubes of six or less than six digits in your exams. Hence, knowledge of the key which contains cubes of numbers from 1 to 10 is more than sufficient. However, since we have dealt with advanced level problems also, you are well equipped to deal with any kind of situation.

COMPARISON

As usual, we will be comparing the normal technique of calculation with the Vedic technique. In the traditional method of calculating cube roots we use prime numbers as divisors.

Prime numbers include numbers like 2, 3, 5, 7, 11, 13 and so on.
Let us say you want to find the cube root of 64. Then, the process of calculating the cube root of 64 is as explained below.

2	64
2	32
2	16
2	8
2	4
2	2
	1

First, we divide the number 64 by 2 and get the answer 32.

• 32 divided by 2 gives 16
• 16 divided by 2 gives 8
• 8 divided by 2 gives 4
• 4 divided by 2 gives 2
• 2 divided by 2 gives 1

(We terminate the division when we obtain 1) Thus, 64 = 2 * 2 * 2 * 2 * 2 * 2
To obtain the cube root, for every three similar numbers we take one number. So, for the first three 2’s, we take one 2 and for the next three 2’s we take one more 2.

When these two 2’s are multiplied with each other we get the answer 4 which is the cube root of 64.

It can be represented as:
64 = 2 * 2 * 2 * 2 * 2 * 2

2*2 equals 4. Hence, 4 is the cube root of 64.
Similarly, to find the cube root of 3375 by the traditional method, we can use the following procedure.

5	3375
5	675
5	135
3	27
3	9
3	3
	1

3375 = 5 * 5 * 5 * 3 * 3 * 3
5*3 = 15

Hence, 15 is the cube root of 3375.

After studying the above two examples, the reader will agree with me that the traditional method is cumbersome and time- consuming compared to the method used by Vedic mathematicians. However, you will be shocked to see the difference between the two methods when we try to calculate the cube root of some complicated number.

Example: Find the cube root of 262144

The Traditional Method      The Vedic Mathematics method

2	262144
2	131072
2	65536
2	32768
2	16384
2	8192
2	4096
2	2048
2	1024
2	512
2	256
2	128
2	64
2	32
2	16
2	8
2	4
2	2
	1

262 144 = |64

= 2 * 2 * 2 * 2 * 2 * 2

= 64

It can be observed from the comparison that not only does the Vedic method help us to find the answer in one line but also helps us to find the answer directly without the need for any intermediate steps. This characteristic of this system helps students in instantly cracking such problems in competitive exams.

With this comparison we terminate this study. Students are urged to solve the practice exercise before proceeding to the next chapter.

EXERCISE

PART A

Q. (1) Find the cube roots of the following numbers with the aid of writing material.
(1) 970299
(2) 658503
(3) 314432
(4) 110592
(5) 46656
(6) 5832
(7) 421875
(8) 1030301

PART B

Q. (2) Find the cube roots of the following numbers without the aid of writing material.

(1) 132651 (5) 474552
(2) 238328 (6) 24389
(3) 250047 (7) 32768
(4) 941192 (8) 9261