It's the question from these threads:

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).

calculation 0

Other two examples of the calculation.

calculation 1

calculation 2

Let's move to the ^3, start with the differences between 5^3 and 6^3

dif1

dif2

calculate version of the differences

difcal

And calculate version from the ^3 table.

cal1

cal10

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.