Is there actually a mathematical rule:

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My daughter (who is typically very good at math) has always been confused about not "distributing" powers into a binomial (x + 2)^2 is (x + 2)(x + 2) not (x^2 + x^2). She knew it shouldn't be done, but for her, if she does not have a reason, she does not fully understand (like the discussion we had yesterday of why 0^0 = 1). I finally ended up telling her this, but I don't know if there is actually something out there that states this.

Everything in math comes down to addition and subtraction. Multiplication and division is just adding and subtracting multiple times. Powers and roots are just multiplication and subtracting several times. They have an order:
1. + and -
2. * and /
3. ^ and roots

You must go in order for these rules to work correctly. It is true that powers and roots are also adding and subtracting several times, but you must arrive there by using multiplication or division. If you try to put a power into a quantity with an addition or subtraction sign, the quantity becomes invalid because the step of multiplication is missing. In the same sense, you can put a power into a monomial in the form of a constant and one or more variables (or just more than one variable) because step 2 is being used, and that is next to step 3. There is no skipping order.

There is a little more to it, but that's the basics of it.

She then related it to: PE MD AS (orders of operations) and asked if that was similar. I told her it was.

Now, when she makes a small mistake like that in her algebra, I just remind her of "Mom's Law." The Mom's Law thing is a joke, but is there actually a rule out there (not just something assumed) that states this?

OK, let's work an example, substituting 3 for x.

(3+2)^2=(3+2)(3+2)=(5)(5)=25.

(3^2+3^2)= (9+9)=18.

The statement in brackets is always worked first, no matter what it is, and the functions inside he brackets are worked in the same order inside the brackets as you have listed here.

True, but she wanted to know *why*. She is always driving her teachers crazy with these questions. They either tell her it does not matter and just accept it (even though she is finishing up Algebra 1, she has the thoughts of someone who is in advanced level calculus, she just doesn't know all the rules to do it yet), or they tell her she will find out later. I think last year was the first time a teacher said that he actually did not remember. She is bothered if she just "accepts" that something is the way it is without understanding why (she is compelled to figure out why). It is like anything to the power of zero is one. She watched a proof on it yesterday (on youtube) which she understood [the steps], but still complained that they did not answer her question.

It's like providing a proof that 1 = 2. Everyone knows that is not correct, but unless you know the rules of mathematics, you may not know why, because the proof (at a glance) looks correct. She is looking for an example why it does not work. Substituting in numbers for variables might help things, but unless she understands that dividing by zero is illegal because you cannot have an amount of nothing, she just *knows* it is wrong (and with her you do have to go beyond telling her that you cannot divide by zero. Her next question would be "Why." Although in her defense, she does know and understand this one).

I don't think there's much you can tell her beyond what you have already. The brackets show that the function of ^2 is being applied to the entity (x+2). As long as things are in brackets everything outside the brackets will treat it as one number, not as two individuals. The closest thing I can think of to your rule is probably the PEMDAS thing (we used to use BODMAS but it's pretty much the same I think).

I generally think of powers (e.g. x^n) as what you get if 1 is multiplied by the number x an n number of times. I think this is just my brain making up rules rather than it being a solid rule, but it makes sense for me.

With most mathematical concepts, you just have to try working a problem another way and not getting a correct solution. You may have to try lots of numbers until you find one that doesn't work.

As for dividing by zero: ask her if she can count by zero. You can count by ones or twos or threes, but if you add zero to zero, what do you get?

To have a number you can divide into another number, you have to be able to count in increments of that number. Because basically all multiplication is about is a shortcut to counting by one number to get to another number.

You don't distribute addition either. 1+(2+3) = 1+2+3 not (1+2)+(1+3) (this could also be said as "addition is not distributive").

And you can't flip the operands for exponentiation: x^n is not n^x (this could also be said as "exponentiation is not commutative").

There are 2 things in math: abstract ideas that are related to each other in certain ways, and how we write those ideas down. Order of operations (which operations we happen to do first) is all about how we write stuff down.

Sometimes it makes a lot of sense how we write stuff, sometimes someone just made something up and it stuck for no real reason.

Whether or not an operation is commutative or distributive is actually about the abstract ideas. Something like that can be proven to be true (or false), and it doesn't matter how you write it, it will still be true (or false).

I appreciate all the responses, but that's not what I was trying to ask. I am sorry if I was not clear. I was awake from getting sick, and couldn't go back to sleep. My head was groggy and I was trying to use the computer to help me fall asleep again.

Alternative perspectives on things are always useful in maths, helping to build intuition.

You can think of (x+2)^2 as equalling the area of a square whose sides are x+2 in length.



So you can think of the area as being (x+2)^2.
Or, equivalently as the sum of the areas of the 4 components i.e. x^2 + 2x + 2x + 4.

As for 0^0, I wouldn't even go there just yet Saying 0^0=1 isn't quite as "true" as saying 1+1=2. In the former case it's more of a convention to say it's 1, but you might want to consider how zero to the power of any positive number is always zero - from that perspective the calculus idea of a "limit" would suggest 0^0=0. The fact is that, as with asking what 1/0 is, just because you can ask a question, it doesn't mean there has to be a simple numerical answer. (Another example - what is the biggest whole number?)

On the idea of "Everything in math comes down to addition and subtraction", it's a good observation that multiplying is just repeated addition and powers are just repeated multiplications. You can take the idea even further to get a very austere kind of system: throw away subtraction because it's just the opposite ("inverse") of addition. Also, even addition isn't the most fundamental kind of operation, because you can think of that as just repeatedly taking the next ("successor") number.

Again, I do thank you for your replies. They give me something to think abut. The only reason I keep pressing at this one point is because this is how my kid thinks. She has been this way since she could talk and started asking questions. She does not want examples and proofs. She wants the basic mechanics.

Let me explain what she wants another way. Let's relate a computer to a calculus problem. Imagine she saw a computer and wanted to know how it worked (maybe a limit problem would be a good example). I could tell her basically that the motherboard has cards, a processor and drives all hooked up to a mother board. )That would be like explaining the different parts of the equation to her by name.) I could further explain how each of these parts work. (Explaining what each individual part of the limit works (lim ... x --> 0 ... (x+0)/(x-2) ... etc)). This would still not satisfy her. She would want to see how they work together (solve the equation - this is where most of you have been leading her). She now sees how the HD is hooked up to the mother board and the different parts of the computer interact with each other, making the computer work. You could even show her that if you remove certain parts, the computer would not function properly. They are designed to work together. (just like each part of a limit problem works together, if you mix the orders up or take something away, the problem will not have the correct solution). Conversely, if you need the computer to do something it normally does not do, you have to add hardware or software (like if you have 0/0, you can still find the limit, but you need to use L'Hospital's R rule) Sometimes, if you remove a part, the computer will not function at all. If you take the CPU out, you may as well not have a computer. (likewise, if you remove the term 'Lim', you do not know what you are solving for. She would realize that the CPU is very important, but there would be more questions. How does it work. Way does it work. Why does it have different speeds depending on the chip? (This would be like asking about the specifics of the function.) She would finally just cut to the chase and ask exactly what the CPU does. In a nutshell, she wants to know that the computer is just calculating using sequences of ones and zeros. That's all a computer really is is computing ones and zeros, using a power source to make the circuitry work . (All mathematics is is addition (and subtraction, or undoing addition)). In the end, what she really wanted to know is how the computer uses a language called binary to run, and then I would explain what binary is, and how to use it. (or even more basic is that the circuits are open or closed, and that's how it works.

If you would talk to her about atoms, you would have to dive down deep into quantum mechanics. She HAS to know why things work the way they do. It is more than knowing when to use the rules, proofs and laws. It is looking deep into the basics of mathematics. Probably deeper than most people do. She is just lucky that I love numbers and have given this a lot of thought myself. My middle daughter does not enjoy math, so I tell her that math is like a game (or a puzzle - she likes these). You have to know all the rules to play the game correctly. Sometimes you can get lucky and not have to use the rule you are unfamiliar with, but if you need it and don't know it, you probably will not win the game