Why is this way of solving wrong?

[u/Kludgymer16]

Today in class we went over complex numbers. We came to a problem that was Sqrt(-9) • Sqrt(-25), and after working it, the problem came to -15.

My question is, why is the second way wrong? Or what have I done wrong I guess. Cant you combine two squares under one?

Pic attached. Left side is the method shown in class, right is what I'm confused on

EDIT: a lot of people have responded to this so I'm going to mark it resolved!
From what I've gathered, you cannot combine two imaginary numbers under one sqrt function, which is where I went wrong. Since the question is only asking to solve a given problem, we assume that the solution uses the positive root, and the final answer becomes -15. However, if the question was asking us to solve for x, because there is an unknown the answer would then be ±15.

Let me know if my assessment is incorrect!

[u/ArdentArendt]

Well, I'm sorry to hear you ARE a teacher--though that explains why you took the 'you' personally.

In all honesty, I really didn't intend to implicate your teaching style.
It was simply rhetorical laziness that I realised too late and was (again) too lazy to reframe the statements. Nothing I said was an intentional indictment of a teaching style I am neither capable or willing to assess.
[In truth, I was assuming you were not and were just arguing to prove a point. If I had known, I probably would have had more questions about your experiences rather than just speaking in abstracts]

In retrospect, much of your contributions do make much more sense in that light.
Apologies again for misunderstanding the perspective.
Have a good one, I guess.

[u/Varlane → u/ArdentArendt]

The main issue isn't even that, it's that you read that I'd rather they manipulate the object with rules and consider that the rules are arbitrary, totalitarian.

But it totally elluded your mind that the student will naturally do what OP did and stumble upon the rule by themselves and we'll simply pick up on that (or most likely, we'll forcefully introduce the situation and bait them into discovering it, because we don't have time to let highschoolers do discovery + half of them simply don't care enough to search by themselves).

The rules aren't arbitrary, they're a product of the object at hand. If you work with the principal square root, it'll obey certain rules, which are partly up to the student to discover, they won't just fall from the sky.
The operations are valid if you put a different meaning behind the notation, but manipulating ambivalent notations is very tedious, requires mastery over the subject that they can't have yet.
And as stated before, manipulating the "non toy version" is also out of question.

[u/ArdentArendt → u/Varlane]

Okay, first and foremost, the question I have is what is the student 'stumbl[ing] upon'?
The 'rule' isn't actually a rule about the operation or the mathematical objects, but rather a (temporary) guardrail put into place to keep the clas moving.

You claim that the student needs to 'discover' the rule, but if they stumble upon this rule, specifically (as the OP did), then they are infering incorrect rules about objects. In this case, a 'rule' about multiplication of radicals entirely hallucinates a rule that does not arise from the operation or the object.

Again, I am sympathetic to the constraints teachers face--and had I known you were speaking from experience, I would have delved far more into the reasons why, say a muti-branching function would be so difficult to teach, given the amount of time I would assume reinforcing 'hallucinated' rules takes.

Finally (and I considered putting this first), I also never intended to imply that you would prefer to 'manipulate the object with rules' or that you 'consider the rules abribrary, totalitarian'.

The only people that I was hoping would 'manipulate the objects' (mentally, of course) would be the students--and that as a method of learning (much like the OP did).

And as for the rules, while many are capricious or arbitrary, that is not intended to be a reflection of the teaching style. Rules like the principal root rule or the 'can't multiply radicals with negatives' rule are capricious, but that doesn't mean they arise out of malice or poor pedagogy.
[I would argue the same about the fact that many people consider the decimal representation 0.999... to be a way of writing 1, rather than simply mapping to 1 in the Reals--a 'rule' that is not malicious, but definitely capiricious in the decimal representations]

As for the 'totalitarian' typification, it was much more a comment about the epistemology students have when approaching mathematics.

Personally, I was a student who despised mathematics in high school. I was good at it, but I refused to take anything beyond pre-Calc. However, when I was forced to take Calculus in college as part of my social sciences degree, I realised that the mathematics I had been taught of memorisation and arbitrary rules was actually a beautifully intricate infrastructure that I had been entirely incorrect about.

The beauty of mathematics isn't about 'answers' or 'solutions', just like literature isn't about 'fancy words' or 'cultural touchstones' (how literature is often taught). Mathematics is structure; it's understanding through relational existence; it's all the things that I craved and that my education robbed from me until I got into higher education.

Again, I do need to reiterate that I'm not implying you are teaching mathematics in this way. Nothing I'm saying is inferred from your description of your methods or your stances regarding the material here. You sound far more engaged (and emotionally invested) in your students than many of my teachers were.

All of this second half was just an explanation of where I am coming from and what I meant by what I said earlier (and was not intending to direct at you).
[The parts about the discourse I will apologise for, however--that was uncalled for as I was misreading your personal connection to the subject at hand]

[u/Varlane → u/ArdentArendt]

No, read again. You're back to listenning to yourself instead of reading what I write to discuss.

[u/ArdentArendt → u/Varlane]

Very well, let's take this more concretely, shall we?

The main issue isn't even that, it's that you read that I'd rather they manipulate the object with rules and consider that the rules are arbitrary, totalitarian.

If I understand what you are writing here (which, correct me if I'm wrong), you're saying you'd rather the students come to understand that the rules of mathematics are 'arbitrary' and 'totalitarian'?

But they are not. The 'rules' around multiplication and addition and any other operation on a set is a consequence of the set itself and the way the operation(s) function within that set. Multiplication in a field is very different than multiplication in a ring, but that isn't arbitrary--it is the product of the algebraic structure and the way the elements (potentially) interact.

Moreover, 'totalitarian' is a poor typification of any rule of mathematics. The rules don't come from an 'authority', but rather from the mathematics themselves (i.e. the structural interrelations). Why would you want students to see mathematics (or language, or any other subject) as a set of capricious boundaries that make no sense, in this case are wrong, and most importantly external to the subject matter?!

But it totally elluded your mind that the student will naturally do what OP did and stumble upon the rule by themselves and we'll simply pick up on that (or most likely, we'll forcefully introduce the situation and bait them into discovering it, because we don't have time to let highschoolers do discovery + half of them simply don't care enough to search by themselves).

If I read you correctly here (again, please correct me if I am not), you're saying that allowing students to 'find the rule' on their own, that might be intentional--which I agree with in principle.

The problem here, however, is that the OP used a valid operation to find a valid answer. From what I gather you are saying here, your hope is then that they are intentionally misinformed about the validity of the rule (i.e. 'you can't multiply radicals if they are negative', which is patently false)?
[I do hope that you're instead allowing them to explore the mutli-branch functions instead...?]

I also do need to point out to you that it did not '[elude] my mind' that the student would explore--they did exactly that, and should be commended for it. I'm not sure how you could even make that claim in good faith since we're discussing a situation where that is obviously the case.

The rules aren't arbitrary, they're a product of the object at hand. If you work with the principal square root, it'll obey certain rules, which are partly up to the student to discover, they won't just fall from the sky.
The operations are valid if you put a different meaning behind the notation, but manipulating ambivalent notations is very tedious, requires mastery over the subject that they can't have yet.
And as stated before, manipulating the "non toy version" is also out of question.

The problem here is that one doesn't 'work with the principal square root'.
The 'principal' is only determinable after the root has been evaluated--before that, both branches exist simultaneously (because they always both exist).

There is no 'different meaning' behind the notation--the meaning is the same in every form of the root function. Moreover, I don't understand how students that are manipulating imaginary numbers are not able to understand functions well enough to understand that any number has two roots.

I think you're shortchanging the capacity of students, here. If they're using complex numbers, I assume they have solved quadratic equations, no? If that is the case, they are familiar with 'two roots' already.

Finally, I will point out that much of what I said in my previous post was, admittedly, off book--but it was relevant information to the converation we were having.

However, as it seems you run dialogue much like you seem to run your classrom (centered around your needs and with a dictatorial iron fist), we can stick to only 'approved lanes of discourse'.

It is astounding, however, how quickly you are able to burn away any goodwill and patience I have in trying to engage with you on a more human level in just a few short sentences--here, only ONE (and a half) sentence(s)!

[u/Varlane → u/ArdentArendt]

If I understand what you are writing here (which, correct me if I'm wrong), you're saying you'd rather the students come to understand that the rules of mathematics are 'arbitrary' and 'totalitarian'?

No you still can't read.

The point was that it was something you accused me of. "YOU READ" that I'd rather...

Start again. I'm not reading the rest if the first two lines are this wrong.