Square of three digit number

Square of three digit number

Consider 3-digit number as abc.
Follow these steps to better understand:
Step 1: get  \(\mathbf{c^{2}}\) by multiplying right side digits [c\( \times \)c]
Step 2: get 2bc by criss-cross product of b and c [(b\( \times \)c)+(b\( \times \)c)=2bc]
Step 3: get 2ac+ \(\mathbf{b^{2}}\) by criss-cross product of a and c, criss cross product of middle digits b and b [(a\( \times \)c)+(a\( \times \)c)+(b\( \times \)b)]
Step 4: get 2ab by criss-cross product of a and b [(a\( \times \)b)+(a\( \times \)b)]
Step 5: get  \(\mathbf{a^{2}}\) by multiplying right side digits [a\( \times \)a]

https://www.hamaraguru.com/assets/files/2018-01-08/1515410685-504019-imgur-5.jpeg


This can be written as:
https://www.hamaraguru.com/assets/files/2018-01-08/1515411786-969356-imgur-6.jpeg

 

Example: \( 713^{2} \)
Step 1: Break the number such that a=7, b=1 and c=3
Step 2: Compute \(a^{2}\) , ab, ac, bc and \(c^{2}\) and write as shown in the updated formula.
Step 3: Notice the second row of the formula is obtained by repeating the second, third and fourth term.
Step 4: Compute \(b^{2}\) and place it in the 3rd column below ac term.
Step 5: Now, add the terms taking care of the carry-forward in each step to obtain the square of the number.

https://www.hamaraguru.com/assets/files/2018-01-08/1515412114-733784-imgur-7.jpeg

So, \( 713^{2} \) = 508369