Question

Find the cube of 99 by Nikhilam formula of Vedic Mathematics.

Answer 1

Hint: Nikhilam’s formula is used to calculate the cube of a number without actually multiplying the number thrice. Nikhilam’s formula for cube is given by,
     \[\text{Cube}={{s}^{2}}\left( \text{number}+\text{2}d \right)/s\times 3{{d}^{2}}/{{d}^{3}}\]
where, s = sub-base.
     d = difference.
Since 99 is a two-digit number, the base of the number is 10.
Sub-base is given by the following formula:
     \[s=\dfrac{\text{number}}{\text{base}}\]
Difference is given by the following formula:
     \[d=\text{number}-\text{sub-base}\times \text{base}\]

Complete step-by-step answer:
First, we find the values of sub-base and difference.
Sub-base is calculated as:
     \[s=\dfrac{\text{number}}{\text{base}}=\dfrac{99}{10}\approx 9\]
Difference is calculated as:
     \[\begin{align}
  & d=\text{number}-\text{sub-base}\times \text{base} \\
 & \text{ }=99-9\times 10 \\
 & \text{ }=99-90 \\
 & \text{ }=9 \\
\end{align}\]
Therefore, cube can be calculated as:
     \[\begin{align}
  & \text{Cube}={{s}^{2}}\left( \text{number}+\text{2}d \right)/s\times 3{{d}^{2}}/{{d}^{3}} \\
 & \text{ }={{\left( 9 \right)}^{2}}\left( 99+2\times 9 \right)/9\times 3{{\left( 9 \right)}^{2}}/{{\left( 9 \right)}^{3}} \\
 & \text{ }=81\left( 99+18 \right)/9\times 3\times 81/729 \\
 & \text{ }=81\times 117/9\times 3\times 81/729 \\
 & \text{ }=9477/2187/729 \\
\end{align}\]
Now, we start from the extreme right. The last digit of 729 becomes the last digit of the result, that is, 9. The remaining, that is 72, is added to 2187 as follows:
     \[\begin{align}
  & \text{cube}=9477/2187/729 \\
 & \text{ }=9477/2187+72/9 \\
 & \text{ }=9477/2259/9 \\
\end{align}\]
To get the second last digit, we look at the middle term. The last digit of 2259 becomes the second last digit of the result, that is 9. The remaining, that is 225, is added to 9477 as follows:
     \[\begin{align}
  & \text{cube}=9477/2259/9 \\
 & \text{ }=9477+225/9/9 \\
 & \text{ }=9702/9/9 \\
\end{align}\]
Therefore, the cube of 99 is 970299.

Note: Apart from knowing the formula, the procedure for finding the cube should also be known properly. ‘/’ sign should not be mistaken as a division sign. We should always simplify the individual terms and only after getting 3 different terms can we start the procedure of transferring and adding.