Vedic Mathematics

Using the Urdhva Tiryakbhyam Sutra, we can multiply the digits of 1111 in a vertical and horizontal manner. The result is 14641.

According to the Nikhilam Sutra, the square of a number can be found by subtracting the sum of the digits from the number itself and then multiplying the result by the sum of the digits. In this case, 108 - (1 + 0 + 8) = 99 and 99 * (1 + 0 + 8) = 11664.

Using the Ekadhikina Purvena Sutra, we can simplify the expression as follows: 12345 + 12346 + 12347 + 12348 + 12349 = (12345 + 12349) + (12346 + 12348) + 12347 = 24694 + 24694 + 12347 = 62025.

The Anurupye Shunyamanyam Sutra states that if the unit digit of a number is 0, then the cube of that number can be found by multiplying the number by itself twice. In this case, 123^3 = 123 * 123 * 123 = 186193.

The Dhruva Bheda Sutra states that the square root of a number can be found by subtracting the difference between the number and the nearest perfect square from the nearest perfect square. In this case, the nearest perfect square to 144 is 121, and the difference between 144 and 121 is 23. Subtracting 23 from 121 gives us 12, which is the square root of 144.

The Vyashtisamutpatti Sutra states that the difference between two numbers can be found by subtracting the smaller number from the larger number. In this case, (12345 + 6789) - (12345 - 6789) = 12345 + 6789 - 12345 + 6789 = 13578.

The Triyambakam Sutra states that the cube of a number can be found by multiplying the number by itself three times. In this case, 125^3 = 125 * 125 * 125 = 1953125.

The Panchajanya Sutra states that the square root of a number can be found by dividing the number by 2 and then adding 1 to the result. In this case, 256 / 2 = 128 and 128 + 1 = 129. Therefore, the square root of 256 is 16.

The Ekadhikena Purvena Sutra states that the sum of a series of consecutive numbers can be found by multiplying the number of terms by the middle term. In this case, there are 6 terms, and the middle term is 12347. Therefore, the sum of the series is 6 * 12347 = 74125.

Using the Urdhva Tiryakbhyam Sutra, we can multiply the digits of 144 in a vertical and horizontal manner. The result is 20736.

The Dhruva Bheda Sutra states that the square root of a number can be found by subtracting the difference between the number and the nearest perfect square from the nearest perfect square. In this case, the nearest perfect square to 625 is 625, and the difference between 625 and 625 is 0. Subtracting 0 from 625 gives us 625, which is the square root of 625.

The Vyashtisamutpatti Sutra states that the difference between two numbers can be found by subtracting the smaller number from the larger number. In this case, (12345 + 6789) - (12345 - 6789) = 12345 + 6789 - 12345 + 6789 = 13578.

The Triyambakam Sutra states that the cube of a number can be found by multiplying the number by itself three times. In this case, 169^3 = 169 * 169 * 169 = 4826809.

The Panchajanya Sutra states that the square root of a number can be found by dividing the number by 2 and then adding 1 to the result. In this case, 400 / 2 = 200 and 200 + 1 = 201. Therefore, the square root of 400 is 20.