Would you elaborate on that? That doesn't make any sense to me
2x3= 2+2+2 AxB= A+A B times.
Raising to a power is repeated multiplication.
2^3= 2x2x2 A^B = AxA B times
Roots are the inverse of powers
Division is the inverse of multiplications.
Given this… the OC’s comment about giving exceptions to powers in the rules is nonsense because they are still operations.
Multiplication is a scaling operation.
In the integers, it is the same thing, with a rule for negative times.
In the rational numbers, you can generalize to adding a number to itself a rational number of times. Adding a/b to itself p/q times is defined as adding a to itself p times, with a divisor that is adding b to itself q times.
So adding the rational 5 (5/1) with itself 2_and-1/3 (4/3) times just means (5+5+5+5) / (1 + 1 + 1) = 20/3, a rational that we can also write as 6 -and-2/3. Only addition of integers is used.
Which is the same as rational multiplication (a/b)(p/q) = (ap)/(b*q).
But it's true that you can't say that rational multiplication is the same as _integer_ repeated addition.
Multiplication also flips signs (direction). That's why a negative * negative = positive, whereas a negative + negative = negative. The addition operator does not flip direction.
Multiplication is not generally the same as multiple addition operations. However, it has the same result for positive whole numbers.
I am not writing this to flippant. I'm a 40-something year old engineer and I litterally thought the same thing until a year ago-ish. I was trying to build a function (I can't recall what exactly) for a project and I said, well instead of multiplying these terms (which wasn't efficient for what I was doing or something), I'll just cycle through and add the terms X-number of times. ChatGPT said "uh, no, that's not the same thing," to which I said NFW and proceeded down a deep rabbit hole on the difference. I'm glad I did because I also now have a much better intuition for imaginary numbers, which I was severely lacking before.
I like it when people who are clearly smart do things like this! Makes me feel better.
Reminds me of this story: https://news.ycombinator.com/item?id=5663709
Learn logarithms.
The comment below by drfloyd51 is similarly wrong ... hahajk didn't say anything about "giving exceptions to powers in the rules".
And the comment by speglt is also wrong:
> If taking a square root is an inherent use of a two, then dividing a number by another number should also be a use of a two, because there are two operands.
The number of operands isn't relevant; the degree of the root is.
And from SideQuark we get
> One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt.
But hahajk said nothing about anything being disqualified. (Not to mention that division is not raising to the -1 power ... raising what? n/d = n * d^-1, but this is completely irrelevant to hahajk's argument.)