Square 104 without long multiplication

Question

Squaring just above 100 is one of the easiest tricks to look impressive with. Solve in your head. What is 104^2? Aim: under 8 seconds.

Accepted Answer

The surplus over 100 is 4. Add it to the number for the front, then append the square of the surplus: 104 + 4 = 108 \Rightarrow 10800, \qquad 4^2 = 16. 104^2 = 10800 + 16 = 10816. So 104^2 = \mathbf{10816}. Check: (100+4)^2 = 10000 + 800 + 16 = 10816. Matches. Takeaway. Near 100: (100+d)^2 = (100 + 2d)	imes 100 + d^2, i.e. "number plus surplus, then d^2." Squaring 101–109 becomes trivial.

Square 104 without long multiplication

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