Question

Find the cube of each of the following numbers using Nikhilam Sutra
I.59
II.96
III.91

Answer 1

Hint: First of all, we divide the given number with a base value example 10, 100 etc. depending on the number of digits in the given number. Then, find the value of sub-base by using the formula, $ s = \dfrac{{number}}{{base}} $ . The difference between the original number and the number obtained from the product of base value and sub-base value is given as - $ d = [original\,number - (value\,of\,subbase \times base) $ and then substitute the values in the formula of cube i.e. $ cube = \left. {{s^2}(No. + 2d)} \right|\left. {s \times 3{d^2}} \right|{d^3} $ to get the answer.

Complete step-by-step answer:
The formula of finding a cube by Nikhilam Sutra is given as,
 $ cube = \left. {{s^2}(No. + 2d)} \right|\left. {s \times 3{d^2}} \right|{d^3} $
Sub base, $ s = \dfrac{{number}}{{base}} $ .
 $ d = [original\,number - (value\,of\,sub - base \times base) $
Now, we will calculate the cube of numbers,

(i.) 59
As 59 is a 2 digit number the base is 10, so sub-base can be given as $ s = \dfrac{{59}}{{10}} = 5.9 \approx 5 $
Difference $ d = [59 - (5 \times 10)] = 9 $
So,
 $
\Rightarrow cube = \left. {{{(5)}^2}(59 + 2(9))} \right|\left. {5 \times 3{{(9)}^2}} \right|{(9)^3} \\
\Rightarrow cube = \left. {25(78)} \right|\left. {5 \times 243} \right|729 \\
\Rightarrow cube = \left. {1950} \right|\left. {1215} \right|729 \;
  $
We take the last digit of 729 that is 9, rest of the two digits are added to the number on the left.
 $ cube = \left. {1950} \right|\left. {1287} \right|9 $
Now we take the last digit of 1287 that is 7 and add the rest of the digits to the number on the left.
 $ cube = \left. {2078} \right|\left. 7 \right|9 $
As all numbers in the second and third columns are in one digit, we can consider this as our final answer.
So cube of 59 is 207879.
So, the correct answer is “ 207879.”.

(ii.) 96
It is a 2 digit number so we take the base as 10, so sub base is $ s = \dfrac{{96}}{{10}} = 9.6 \approx 9 $
Difference $ d = [96 - (9 \times 10)] = 6 $
 $
\Rightarrow cube = \left. {{{(9)}^2}(96 + 2(6))} \right|\left. {9 \times 3{{(6)}^2}} \right|{(6)^3} \\
\Rightarrow cube = \left. {8748} \right|\left. {972} \right|216 \\
  $
Follow the same procedures as in part (a)
 $
\Rightarrow cube = \left. {8748} \right|\left. {993} \right|6 \\
\Rightarrow cube = \left. {8847} \right|\left. 3 \right|6 \;
  $
Cube of 96 is 884736.
So, the correct answer is “ 884736”.

(iii.) 91
It is a 2 digit number so base is 10 and sub base is $ s = \dfrac{{91}}{{10}} = 9.1 \approx 9 $
Difference $ d = [91 - (10 \times 9)] = 1 $
 $
\Rightarrow cube = \left. {{{(9)}^2}(91 + 2(1))} \right|\left. {9 \times 3{{(1)}^2}} \right|{(1)^3} \\
\Rightarrow cube = \left. {8463} \right|\left. {27} \right|1 \\
\Rightarrow cube = \left. {8465} \right|\left. 7 \right|1 \;
  $
Cube of 91 is 846571.
So, the correct answer is “846571”.

Note: Students might misinterpret the formula of finding a cube using Vedic math’s as division, the terms are just separated in columns, the symbol doesn’t represent division. Don’t get confused otherwise you may get the wrong answer. So, students must follow Vedic math’s method before solving such questions.