Cubes by Bhārati Kṛṣṇa Tīrtha

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Here we will be learning how to find the cubes of two digit numbers. We will be using the sūtra[1]

आनुरूप्येण

" Ānurūpyeṇa "

" Proportionally "

Steps

The detailed steps are explained below .

Let a and b are the two digits .

Step 1 : We will be writing the four numbers in geometric ratio in exact proportion.

First term will be , common ratio =

second term will be

third term will be

fourth term will be

First term Middle terms (second, third terms) Fourth term
Step 1
Step 2 double of the middle

terms

double of the middle

terms

Step 3 add the two middle terms

add the two middle terms

Final Step

This is nothing but the expansion of the formula for where a and b are two individual digits of the number. This will makes the calculation of cubes very fast and easy. This will be explained through the below examples.

Cubes of natural numbers
1 2 3 4 5 6 7 8 9
1 8 27 64 125 216 343 512 729

Example: 233

Here a = 2 , b = 3 , Using the final step mentioned above.

23 = 8 3 X ( 22 X 3) = 3 (4 X 3) 3 X (2 X 32) = 3 X (2 X 9) 33 = 27
8 36 54 27
8 36 54 Put 7 and carry over 2
8 36 54 + Carry over (2) 7
8 36 56 7
8 36 Put 6 and carry over 5 7
8 36 + Carry over (5) 6 7
8 41 6 7
8 Put 1 and carry over 4 6 7
8 + Carry over (4) 1 6 7
12 1 6 7

Answer : 233 = 12167

Example: 123

Here a = 1 , b = 2 , Using the final step mentioned above.

13 = 1 3 X ( 12 X 2) = 3 (1 X 2) 3 X (1 X 22) = 3 X (1 X 4) 23 = 8
1 6 12 8
1 6 Put 2 and carry over 1 8
1 6 + Carry over (1) 2 8
1 7 2 8

Answer : 123 = 1728

Example: 253

Here a = 2, b = 5 , Using the final step mentioned above.

23 = 8 3 X ( 22 X 5) = 3 (4 X 5) 3 X (2 X 52) = 3 X (2 X 25) 53 = 125
8 60 150 125
8 60 150 Put 5 and carry over 12
8 60 150 + Carry over (12) 5
8 60 162 5
8 60 Put 2 and carry over 16 5
8 60 + Carry over (16) 2 5
8 76 2 5
8 Put 6 and carry over 7 2 5
8 + Carry over (7) 6 2 5
15 6 2 5

Answer : 253 = 15625

See Also

घन - भारती कृष्ण तीर्थ

References

  1. Singhal, Vandana (2007). Vedic Mathematics For All Ages - A Beginners' Guide. Delhi: Motilal Banarsidass. pp. 237–242. ISBN 978-81-208-3230-5.