Adding exponents/thorough way.
What is the long way to add any exponents. AKA an alternative to literally adding say 2^2 + 5^2. I think I've seen egyptian math that does stuff with grids that use principles that would be useful but I don't know what they are anymore and I'm trying to find a formula base as well as basic patterns to use or whatever else is equivalent(literally and in the broadest sense.). I'm looking to try to build as many alternative forms and find patterns to do it with but I"m struggling a lot(my base reason is to find a quicker way for basic operations in my head but I always want to try to take them farther and cross reference to other areas of math, and find as many alternative ways to do something, etc). Any hints? I couldn't find a way to describe it from only x so I'm stuck and got frustrated with the number line method I was doing looking bluntly for the patterns to find principles of the change for a quick practical method.
Also I put this here becuase I'm not sure where it fits. I'm not sure what constitutes things like number theory etc so I don't know what this falls under. And my viewpoints for all math are to learn it in every way possible. But here I"m still starting out. As this may indicate I'm not that far yet. I've just now caught up with where I was in high school 10 years ago in college math. But math is my passion in life so I always work on it regardless where I can.
Edit: I'm looking for both x^2+y^2 and multiple exponents values x^a + y^b. like I said I'm trying to figure out principles and formulas at least. Side note I have health problems so I'm a little slower that I might be other wise. it affects me mentally slightly. So If it seems stupid where I'm asking from it's because I'm getting back into practice with math and it takes me longer right now for health reasons.
BTW any help for any part of it is helpful. this is a broader subject(maybe). 8) Thank you for any help. I like looking at everything. So help with just a focused part is also good not just the broadest aspect. I like finding all ways. But I'm a little bit debilitated atm.
One method I was using was number lines where I took basically 1 - 10 then above the x^2 version then using that to find patterns but came up with a longer formula(x^2 + (x +(y-x)^2) not sure how to reduce to x if even possible or if it is correct.). Having difficulty. The difference of those is the odd number pattern btw. 2x(+-)1. I was using it for both formula stuff and trying to find the simple difference between values and find the pattern but I'm getting mentally drained to quick and thought I would ask for help. If you take the differences and products of the exponential forms there is almost seemingly a patter of some sort.(there must be or it wouldn't exist). but I can't get my mind around it.
x: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
x^2: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
(x+1)^2 - x^2(odd numberpattern): 3, 5, 7, 9, 11, 13, 15, 17, 19
(x+1)^2 + x^2: 5, 13 25, 41, 61, 85, 113, 145, 181
difference of above: 8 12, 16, 20, 24, 28, 32, 36 (+4 each difference)
I doubt that shows up well. I need a grid for it. 8) direct things are straight below like x^2. Others that are from two numbers are between or are supposed to be.
Edit2ish?!: I found this but it doesn't save time. 8)
((x-y)^2 + (x+y)^2)/2 = x^2 + y^2 AKA the difference and the product each squared divided by two is the same as the individual numbers squared and added. I think I found this along time a go but it's been driving me batty since I couldn't remember. Anyone know one that is better mentally? This is the same work as the original or more, but it might work great on a grid(I think lol).