Why does my answer to 80.25^2 not match what the calculator outputs, even though my math seems sound?

Question

When I input 80.25^2 into a calculator, the answer it outputs is 6440.0625.

But when I do it by hand, I get a result of 6440.0725.

The only discrepancy I can think of that would cause this is an issue carrying the one when multiplying 5 and 0 early in the process.

When you multiply 5 by 0, the answer is 0. That part I am very clear on. But there is a carried one from the prior step, multiplying 5 by 2.

When you're multiplying by zero, is the answer zero, even when you have a carried number in that column? I always assumed that the carried number would still be included, making the answer nonzero. But maybe I've just never ran into this exact circumstance before.

Answer 1

80.25

= 80+ 1/4

(80 + 1/4)^2

use the formula (a+b)^2 = a^2 +2ab+b^2 with a= 80, b= 1/4

= 6400 + (2)(1/4)(80) + 1/16

= 6400 +40 + (1/2)(.125)

= 6440 + 0.0625

= 6440.0625

80.25

80.25

40125

16050

642000

6440.0625

you didn't line them up right, similar to the misaligned 3 partial multiplications above

making you add 5+2 =7 instead of 5+1 = 6