This document provides methods for deriving cube roots of exact cubes. It explains: - The values of the digits 1-9 when cubed - How to group the digits of a number and determine the number of digits (N) in the answer - How to find the leftmost (L) and rightmost (R) digits using the grouped digits and cubed values - An additional middle (M) digit is needed for numbers with three digits in the answer - Examples are given to show finding L, R, and M to derive the cube root - Later examples show that determining M can sometimes have multiple possible values, making the method ambiguous

1. Cube roots of exact cubes - Part I
We know that in our exams almost there may be two to three questions of
derivation of cube roots of exact cubes. I think, all of you know the traditional
methods but these methods are quite cumbersome and time consuming. So, here
we will be learning tricky methods to derive cube roots of exactcubes.
Be with me and have patience.
First of all let us remember followings by heart:-
13 = 1 So, 1 = 1
23 = 8 So, 2 = 8 (10-2)
33 = 27 So, 3 = 7 (10-3)
43 = 64 So, 4 = 4
53 = 125 So, 5 = 5
63 = 216 So, 6 = 6
73 = 343 So, 7 = 3 (10-7)
83 = 512 So, 8 = 2 (10-8)
93 = 729 So, 9 = 9
From here onwards, we will call
The first digit of the answer form left = L****
The last digit of the answer = L****R
Numbers of digits in the answer = N
Let me explain it:- (let us go by an example)
Q. Derive Cube Root of ABCDEFGH.
Ans :- Steps are as follows:-
(a) Make groups of digits from right. i.e. AB, CDE, FGH
So, numbers of digits in answer = 03 (N = 3)
(b) To find the left digit i.e. L, see the most left group i.e. AB in the Question.

2. This ‘AB’ will be definitely just more than the cube of digits from 1 to 9. For
example if ‘AB’ = 56, it is just more than 33 = 27 but less than 43 = 64
So, L = 3
(c) To find the right digit of the answer, you have to just see the most right digit. It
will be definitely the last digit of cube of the numbers from 1 to 9.
So, R = Now known from 1 to 9
So, up to 13
= 1 to 993
= 970299, we can derive cube roots very easily using the
values of L and R digits. Here N = 2. Mind againhere that we are talking about
exact cubes only.
Let us see some examples:-
Q1. Cube root of 46656
Ans – Make groups first. It will be 46, 656 So, N = 2
R = 6 (Because 6×6×6 = 216)
L = 3 (Because 33 < 46 > 43)
So, the cube root = 36
Q2. Cube root of 704969
Ans – Make groups first. It will be 704, 969
R = 9 (Because 93 = 729)
L = 8 (Because 83 < 704 > 93)
So, the cube root = 89
Q3. Cube root of 804357
Ans – 804, 357 = 93 (I have collapsed all the above-mentioned steps into one
line)
So, gentlemen, are u confident now? If yes, let us proceedfurther or else revise
it once more!
Cube roots of exact cubes - Part II
So, gentlemen, u are confident now.

3. Now, we will see the cube roots of (100)3
to (999)3
. So, obviously we will have
answers of three digits. Here N = 3. Form the previous section, we know to derive
L , R and N. Here again you have to apply these apart from a middle term. Let us
call it M hereafter.
Now let us go step by step to find the value of M using examples:-
Q1. Cube root of 11697083
Ans – Group it first. – 11, 697, 083
So, R = 7 and L = 2 but M = ?
Step I – R3
= 343 (Subtract it from the sum in the
question) 11,697,083 – 343 = 11,696,740 (Remember here
that the most right digit will be ZERO only. Ignore it. Then the
most right digit = 4. You have to mind that you should not
subtract entirely. You have to just find out the most right digit.
That’s all. You can do it even seeing it visually.)
Step II – 3 × R2
× M = 3 × 49 × M = 7M (I have not done 3 ×
49. No need of it actually. Just see the most right digit. It is 7.
Now you have to think a bit that what should the value of M to
get 7M = 4 (4 of step – I) Yes. It is 2. 7 × 2 = 4. So, you have
got the value of M. M = 2.
Ans – 227
Note – In step I, we have just subtracted R3
. Find the most right digit ignoring
ZERO. What if, again you get ZERO. For example – If in the above example
the sum after subtraction is 11, 696, 700. You have to ignore only first ZERO.
For step II you will refer only ZERO.
Q2. Cube root of 103161709
Ans – Group it. 103, 161, 709 So, N = 3, L = 4 and R = 9 but M = ?
Step – I 93
= 729. Subtract it. Got 8 as the most right digit after ignoring ZERO.
Step – II 3 × 92
× M = 3M So, to get 8, M should be 6. M = 6

4. So, your answer is 469.
Go and get a cup of tea and refresh yourself.
Q3. Cube root of 2000376
Ans – Group it. 2, 000, 376 So, R = 6, L = 1, and M = ? (N = 3)
Step – I 63
= 216. Subtract it. You got 6 after ignoring ZERO.
Step – II 3 × 62
× M = 8M So, in order to get 6, M should be 2 or 7. What to do
now? What will be the value of M, 2 or 7. My friends, our solution has taken U-
turn like our T.V. Serials. M is confusing. Call director of the serial. So, leave it
here only. We cannot gamble in our exam. (Do not worry, I am with you.)
Q4. Cube root of 335702375
Ans – Group it. 335, 702, 375 So, L = 6, R = 5 and M = ? (N = 3)
Step– I Subtract R3
. i.e. 125. You get 5 as the most right digit after ignoring ZERO.
Step – II 3 × R2
× M = 5M So, in order to get 5, what will be the value of M? Here,
M = 1 or 3 or 5 or 9 or 7. Friends the problem has now became more critical.
Let us call doctor in spite of director.