It's the question from these threads:
Asking for suggestions about square numbers
Don suggested my trying to explain it more clearly because my scribbles were too messy
Please allow me to continue here because I can't add more photos in the old post T_T I know this is inappropriate manner, but I don't know what else to do... It won't happen again after this.
(Let's continue)
I can't explain this one... but it's a way to calculate. I found it out from processes mentioned above. (Please ignore the green 11^2 part).
Other two examples of the calculation.
Let's move to the ^3, start with the differences between 5^3 and 6^3
calculate version of the differences
And calculate version from the ^3 table.
and that's all for as far as I wrote them down... Got to go back to work since I have been slacking off for 2 days T-T
It's that I found such stuff fun and I want to learn more about it. Thank you JMoravitz and J.G., who kindly suggested some some information. I think I'm too new to math +_+; I need to learn the meaning of the symbols first .. They're like outerspace language to me now. But I'll try! Other suggestions (maybe in basic math first) are very welcome.
You can derive all these results by substituting suitable values of $x,\,y$ in$$(x+y)^2=x^2+y^2+2xy,\,(x+y)^3=x^3+3x^2y+3xy^2+y^3,$$special cases of the binomial theorem.