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### Exploring The 16 Sutras: Advanced Mathematical Examples and Techniques ###

Dive into the world of The 16 Sutras with comprehensive examples that illustrate advanced mathematical concepts. This guide covers various operations such as division, solving equations, and polynomial manipulation. Detailed illustrations explain how to tackle complex problems, making it an invaluable resource for students and educators alike. Understand the relationships between variables and coefficients through practical examples, enhancing your grasp of mathematical principles and techniques. Perfect for anyone looking to deepen their knowledge of sutras and mathematical problem-solving. ###

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### Exploring The 16 Sutras: Advanced Mathematical Examples and Techniques ###

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  1. “The 16 Sutras”

  2. Examples: 1) 252  = 2 X (2+1) / 25 = 625 2) 352 = 3 X (3+1) /25 = 3 X 4/  25 = 1225 3) 1352 = 13 X 14/25 = 18225

  3. Number Base 14 10      8 10 10 10 97 100 112 100 993 1000 1011 1000

  4. Example: 23                               x   13                          ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯                          2 : 6 + 3 : 9  =  299

  5. Example: Divide 1225 by 12.  12         122      5

  6. Example: Divide 1225 by 12.  12         122      5            - 2                    ¯¯¯¯¯    

  7. Example: Divide 1225 by 12.  12         122      5          - 2        -2               ¯¯¯¯¯     ¯¯¯¯    

  8. Example: Divide 1225 by 12.  12         122      5            - 2         -2           ¯¯¯¯     ¯¯¯¯                 10     

  9. Example: Divide 1225 by 12.  12         122      5            - 2         -20              ¯¯¯¯¯     ¯¯¯¯                 10     

  10. Example: Divide 1225 by 12.  12         122      5            - 2         -20              ¯¯¯¯¯     ¯¯¯¯                 102       

  11. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102       

  12. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1

  13. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore:

  14. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12

  15. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 =

  16. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102

  17. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102 Remainder =

  18. Example: Divide 1225 by 12.  12         122      5            - 2         -20     - 4          ¯¯¯¯¯     ¯¯¯¯¯¯¯¯¯¯                 102      1 Therefore: 1225/12 = 102 Remainder = 1

  19. Example 2:      5(x+1) = 3(x+1) Solution: ( x + 1 )              x + 1 = 0     gives     x = -1

  20. Example :                      3x +  7y = 2                       4x + 21y = 6 Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is same as the ratio of independent terms i.e., 2 : 6 i.e., 1 : 3. Hence the other variable x = 0 and 7y = 2 or 21y = 6 gives y = 2 / 7.

  21. Example 1:                            45x – 23y = 113                            23x – 45y = 91 add them,                i.e., ( 45x – 23y ) + ( 23x – 45y ) = 113 + 91                i.e., 68x – 68y = 204         x – y = 3subtract one from other,                i.e., ( 45x – 23y ) – ( 23x – 45y ) = 113 – 91                i.e., 22x + 22y = 22           x + y = 1      and repeat the same sutra, we get x = 2 and y = -1

  22. Example 2:     x3 + 8x2 + 17x + 10 = 0         We know ( x + 3 )3 = x3 + 9x2 + 27x + 27       So adding on the both sides, the term ( x2 + 10x + 17 ), we get             x3 + 8x2 + 17x + x2 + 10x + 17 = x2 + 10x + 17     i.e.,, x3 + 9x2 + 27x + 27 = x2 + 6x + 9 + 4x + 8                  i.e.,, ( x + 3 )3 = ( x + 3 )2 + 4 ( x + 3 ) – 4                      y3 = y2 + 4y – 4 for y = x + 3                         y = 1, 2, -2. •     Hence x = -2, -1, -5         .

  23. Example: 123Step 1: Subtract the nearest power of ten from the number: 12 - 10 = 2Step 2: Double this number and add the number being cubed: (2 x 2) + 12 = 16Step 3: Subtract from this number the power of ten from step one: 16 - 10 = 6Step 4: Multiply this number by the answer in step one: 2 x 6 = 12Step 5: Cube the answer in step one: 23 = 8Step 6: Since we're cubing a 2 digit number, add two zeros to the answer in step two: 1600Step 7: Add 1 zero to the answer in step four: 120Step 8: Add the answers in steps seven and eight to the answer in step five: 1600 + 120 + 8 = 1728Thus, 123 = 1728

  24. Example :  3x2 + 7xy + 2y2 + 11xz + 7yz + 6z2. Step (i):   Eliminate z and retain x, y; factorize                3x2 + 7xy + 2y2 = (3x + y) (x + 2y) Step (ii):   Eliminate y and retain x, z; factorize                3x2 + 11xz + 6z2 = (3x + 2z) (x + 3z) Step (iii):   Fill the gaps, the given expression                  = (3x + y + 2z) (x + 2y + 3z)

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