Yavadunam Tavadunikritya Varga Yojayet

Yavadunam Tavadunikritya Varga Yojayet:
Whatever the Deficiency Subtract that Deficit from the Number and Write Along Side the Square of that Deficit

Yavadunam Tavadunikritya Varga Yojayet is the corollary to Sankalana - Vyavakalanabhyam, the seventh sutra of Vedic mathematics.

This sutra, similar to Anurupyena and Ekadhikina Purvena, can be used to find the square or the cube of a number near base powers of 10, such as 99, 992, 9997, etc.

Squaring a number slightly lower than a power of 10:
Example: 996²
Subtract the number from the nearest power of ten:
1000 - 996 = 4
Subtract this deficit from the number: 996 - 4 = 992
Since the nearest power of ten is 1000, add three zeroes to this result: 992,000
Square the deficit: 4 x 4 = 16
Add this to 992,000 to get the answer
996² = 992,016

Squaring a number slightly higher than a power of 10:
Example: 10025²
Subtract the nearest power of 10 from the number:
10025 - 10000 = 25
Add this surplus to the number: 10025 + 25 = 10050
Since the nearest power of ten is 10000, add four zeroes to this result: 100,500,000
Square the surplus: 25 x 25 = 625 (use Ekadhikina Purvena to quickly square any number ending in 5)
Add this to 100,500,000 to get the answer
10025² = 100,500,625

Cubing a number slightly lower than a power of 10:
Example: 998³
Subtract the nearest power of 10 from the number to find the deficit: 998 - 1000 = -2
Double the deficit and add it to the number: 998 + (2 x -2) = 994
Since the power of 10 we are working from is 1000, add three zeroes to 994 to get our temporary answer: 994000
Subtract our power of 10 from 994: 994 - 1000 = -6
multiply this number by the deficit in the first step: -2 x -6 = 12
Add this number (12) to our temporary answer: 994012
Then add three more zeroes: 994012000
Cube the deficit from the first step: -2³ = -8
Add this to our temporary answer to arrive at our final answer: 99401200 + -8 = 994011992

Cubing a number slightly higher than a power of 10:
Example: 107³
Subtract the nearest power of 10 to find the surplus: 107 - 100 = 7
Double the surplus and add it to the number: 107 + (2 x 7) = 121 Because the power of 10 we are working from is 100, add two zeroes to this number for our temporary answer: 12100
Subtract from 121 the power of ten from the first step: 121 - 100 = 21
Multiply this number by the surplus in the first step: 21 x 7 = 147
Add this number to our temporary answer: 12100 + 147 = 12247
Add two more zeroes to the temporary answer: 1224700 Cube the surplus from the first step: 7³ = 343
Add this to the temporary answer to arrive at the final answer: 1224700 + 343 = 1225043

This sutra can also be applied to numbers that are near a multiple of a power of ten, such as 402, which is near (4 x 100). To learn more about Yavadunam Tavadunikritya Varga Yojayet, check out the resources below.

RESOURCES:
Vedic Mathematics by Sri Bharati Krisna Tirthaji
http://www.vedamu.org/Mathematics/course.asp
http://www.sanalnair.org/articles/vedmath/intro.htm
http://www.vedicganita.org/ganitsutras.htm
http://hinduism.about.com/library/weekly/aa062901a.htm
http://www.vedicmaths.org/
http://www.hinduism.co.za/vedic.htm
Mathemagics by Arthur Benjamin and Michael B. Shermer
http://en.wikipedia.org/wiki/Vedic_math
http://www.tifr.res.in/~vahia/dani-vmsm.pdf
http://www.sacw.net/DC/CommunalismCollection/ArticlesArchive/NoVedic.html