So b * c, which is a product of the variables b and c

Nope. bc is the product of b and c. bxc is Multiplication of the 2 Terms b and c.

according to this textbook

Says person who clearly didn't read more than 2 sentences out of it 🙄

none of the examples on this particular page feature the multiplication symbol ×

and why do you think that is? Do explain. We're all waiting 😂 Spoiler alert: if you had read more than 2 sentences you would know why

That means that the expression bc is just another way of writing b×c;

No it doesn't. it means bxc is Multiplication, and bc is the product 🙄 Again you would've already known this is you had read more than 2 sentences of the book.

it is treated the same other than requiring fewer strokes of the pen

No it isn't, and again you would already know this if you had read more than 2 sentences. If a=2 and b=3, then...

1/ab=1/(2x3)=1/6

1/axb=1/2x3=3/2

this is just a custom

Nope, an actual rule of Maths. If you meant 1/axb, but wrote 1/ab, you've gonna get a different answer 🙄

That should clear up your confusion in interpreting this textbook

says person who only read 2 sentences out of it 🙄

though really, the language is clear:

It sure is when the read the rest of the page 🙄

you don’t dispute that b×c - or b * c - are products, do you

What don't you understand about only ab is the product of a and b?

Elsewhere in this thread you are clearly confused about what brackets mean

Not me, must be you! 😂

They are explained on page 20 of your textbook, where it says that you evaluate the expression inside the (innermost) brackets before doing anything else.

Until all brackets have been removed. on the very next page. 🙄 See what happens when you read more than 2 sentences out of a textbook? Who would've thought you need to read more than 2 sentences! 😂

the “distributive law” is not mentioned, because the distributive law has nothing to do with brackets

And yet, right there on Page 21, they Distribute in the last step of removing Brackets, 🙄 5(17)=85, and throughout the whole rest of the book they write Products in that form, a(b) (or just ab as the case may be).

is not an operation

Brackets aren't an operator, they are grouping symbols, and solving grouping symbols is done in the first 2 steps of order of operations, then we solve the operators.

Thus the expression 3 × (2 + 4) can be evaluated by first performing the summation inside the brackets to get 3 × 6 and then the product to get 1

3x6 isn't a Product, it's a Multiplication, done in the Multiplication step of order of operations.

The textbook then says that it is customary to omit the multiplication symbol and instead write 3(2+4)

It says you omit the multiplication sign if it's a Product, and 3x6 is not a Product. I'm not sure how many times you need to be told that 🙄

again indicating that these expressions are merely different ways of writing the same thing

Nope, completely different giving different answers

1/3x(2+4)=1/3x6=6/3=2

1/3(2+4)=1/3(6)=1/18

You have suggested that you must evaluate this as (2a+2b)² because you must “do brackets first”

Yep

this is not what “doing brackets” means.

Yes it is! 😂

Not what is outside the brackets.

Yes it is! 😂 Until all Brackets have been removed, which they can't be if you haven't Distributed yet. Again, last step of the working out...

Distributing 2 over a+b is not “doing brackets”;

Yes it is! 😂 Until all Brackets have been removed

it is multiplication and comes afterwards

Nope, it's Distribution, done in the Brackets step, before doing anything else, as per Page 21

following your textbook’s instruction to do what is inside the brackets first, this is equal to 2(4)²

Which, when you finish doing the brackets, is 8²

The next highest-priority operation is the exponent

After you have finished the Brackets 🙄

giving us 2×16

Nope. Giving us 8²=64

we now must write the × because it is an expression purely in numerals

Nope! If you write it at all, which you don't actually need to (the textbook never does), then you write (2x4)², per The Distributive Law, where you cannot remove the brackets if you haven't Distributed yet. There's no such rule as the one you just made up

The fact that these two answers are different is because

You disobeyed The Distributive Law in the second case, and the fact that you got a different answer should've been a clue to you that you did it wrong 🙄

what it means to “do brackets” and the distributive law are wrong

No, that would be your understanding is wrong, the person who only read 2 sentences 🙄 I'm not sure what you think the rest of the chapter is about.

Since I’m working off the textbook you gave

Says person who only read 2 sentences out of it 🙄

I referred liberally to things in that textbook

Yep, ignoring all the parts that prove you are wrong 🙄

I’m sure if you still disagree you will be able to back up your interpretations with reference to it

Exact same reference! 😂

it does rather seem like this rule is one established not by the fundamental laws of mathematics but by agreement as they say

You know Mathematicians tend to agree when something has been proven, right? 😂

Care to comment?

Yep, read the whole chapter 🙄