moriquende

I've read everything you've posted, but the problem is you're interpreting the texts in such a way that they support your flawed argument, conveniently ignoring what they're actually saying, such as "if" statements.

Even this textbook that you yourself posted goes against what you're saying if you just bother to look at it outside of your tunnel vision:

Notice something?

Lmao citing yourself and assuming you're correct and smarter than everyone who programs solvers, even those who are known to be respectable and used extensively in academia. Nothing's been established cause you've cited sources that don't support your argument, and repeating them again and again won't make it different. Good day bro, continuing this is useless.

Ok bro now find an expression solver that verifies your solution. I tried Wolfram Alpha, Google, and others, and they all return 128. So either you're wrong, or all people who make these tools professionally are wrong. Not trying to be offensive, but I know where I'm putting my money.

To be clear, the reason you're wrong is because distribution is not part of the brackets step. Brackets are solved before exponents, resulting in 2(8)². Remove the brackets and then it's 2*8²

Exponents come after brackets, so I'm curious to see how you solve that with your logic lol. It has an obvious correct solution, which is 128, but you need to distribute in the brackets step, which comes before exponents, so let's see what you do with it lmao.

Ok bro so answer my question what's the result of the expression I wrote above?

Bro directly after what you underlined it says "if you want to remove the brackets" lol. Selective reading much? "If" means optional. You are free to solve what's inside the brackets first, before multiplying it with what's outside.

Also, the link I posted is literally titled "distributive law", not property. You realize a law can have conditions, right?

Quick quiz for you: what's the result of 2(3+5)² ?

You must not distribute brother, lay it to rest lmao. It's optional. Google distributive law and find me one source saying it's imperative to distribute - there's none. You can even confirm this is true yourself with simple examples like the ones I've mentioned above.

Here is math for kids https://www.mathsisfun.com/definitions/distributive-law.html

Distributive law means you are allowed to distribute, not that you must distribute. I'm so sorry for the amount of effort you're futilely putting into this lmao. Nowhere in all your sources and screenshots is it stated you must distribute, and thus the entire argument breaks down.

Nowhere in your "proof" screenshots does it say anything about distribution being part of the brackets step. Distribution is a method that can help solve equations, but it isn't required. If you have 2(3+5) you're free to solve it as 2*3+2*5 or as 2*8, whichever is easiest. That is because juxtaposition means multiplication and nothing else.

Math textbooks almost universally will either use clear brackets or simply write divisions in 2 lines, which avoids the confusion altogether.

Yes, the math textbook says exactly what I said, that it's a multiplication. There's no mention of it being a separate operation taking precedence. The parentheses in your example are added for clarity.

Whether you give priority to juxtapositions is an open debate with the consensus being to just use parenthesis around when writing in a single line to avoid confusion. However, there is no distribution step taking precedence, as you mentioned, and the whole debate centers around whether the writer was too lazy to add parenthesis.

1/a(b+c)=1/(ab+ac)

Nope, that's wrong. See https://www.wolframalpha.com/input?i=10%2F2%282%2B3%29 for reference.

But factorised terms are multiplications, so they're still following the same rules: a(b+c) = a*(b+c)

Example: 2(3+5)=16, and also 2*3+2*5=16