Anything multiplied by 0 is 0 right so why does x number raised to the power of 0 = 1? isnt it x^0 = x*0 (im turning grade 10 and i asked my teacher about this he told me its because its just what he was taught 💀)
In: Mathematics
The pure mathematician will tell you: It analytically respects the property *a*^(*x*+*y*) = *a*^(*x*) × *a*^(*y*) that holds true when *x* and *y* are positive integers. It is by extrapolation of this property over all other numbers that we get things such as *a*^0 = 1 for all *a* except 0, *a*^(−*x*) = 1÷*a*^(*x*), *a*^(½) = √*a*, and e^(iπ) = −1.
3Blue1Brown (Grant Sanderson) has a video that tackles this question head-on with an appeal to group theory: https://youtu.be/mvmuCPvRoWQ
This….
You won’t get a better explanation
Think of all possible exponents and where zero fits in. If you question any of the other values, sort that out first, since they are easier.
| 3 ^ 2 |9|
|:-|:-|
|**3 ^ 1**|**3**|
|**3 ^ 0.5**|**1.732 …**|
|**3 ^ 0.1**|**1.116 …**|
|**3 ^ 0**|**1**|
|**3 ^ -0.1**|**0.896 …**|
|**3 ^ -0.5**|**0.577 …**|
|**3 ^ -1** |**0.333 …**|
|**3 ^ -2**|**0.111 …**|
Here is a second pass at why the question points to a deep puzzle, I think.
The **usual explanation** for why x^0 = 1 points to something like the relationship between x^m and x^(m-1), extrapolating from there. So, e.g., if 10^3 = 1,000 and 10^ 2 = 100, we see that reducing the exponent by 1 is done by dividing by the base once. And once we notice that, we get a tidy path to 10^0, which is 10^1 / 10, and so on well into the negative numbers.
**But** notice that exponents are often defined in very similar, algorithmic terms:
“The exponent of a number says how many times to use that number in a multiplication.” [https://www.mathsisfun.com/definitions/exponent.html](https://www.mathsisfun.com/definitions/exponent.html)
“An exponent refers to the number of times a number is multiplied by itself.” [http://www.mclph.umn.edu/mathrefresh/exponents.html](http://www.mclph.umn.edu/mathrefresh/exponents.html)
“An exponent refers to how many times a number is multiplied by itself.” [https://www.turito.com/learn/math/exponent](https://www.turito.com/learn/math/exponent)
Of course, on those definitions, answers to things like n^0, n^-1, and n^e are very puzzling! The instruction “multiply 5 by itself never” does not seem to lead to 1. Imagine I asked “What is the difference between no numbers?”. Our most familiar arithmetic operators need some operands! And why would we get a math answer from *not* doing any math (which is what the ordinary definition suggests we should do in the x^0 cases).
So what to do about that puzzling? Well, as the wikipedia for exponentiation, [https://en.wikipedia.org/wiki/Exponentiation](https://en.wikipedia.org/wiki/Exponentiation), explains, one way to figure the rest is to start to use properties of the natural number exponents and extrapolate from there. That’s great (and gets us something like the usual explanation). But it leaves our original understandings of what an exponent is high and dry. Which might be why the high schooler’s math teacher balked.
What to do then? One answer is to appeal to **special cases**, as the University of Minnesota link above does. But we still need an explanation for why we want to admit those special cases. Especially because this means that our definition of exponentiation is now branched or disjunctive.
As far as I know, there are two related answers:
1. A pragmatic answer. Having the disjunctive definition allows us to do more math more easily. Undefined bits gum up the works. We’d like to do things like figure out how to make sense of (x^m)*(x^n), and if we have to start adding all sorts of qualifications, that’s really going to undermine how **usable** math is.
2. A deeper principled answer. It turns out that the original, familiar definition was overly simple. There is a deeper, principled, and **unifying** definition of exponentiation which explains all of the ostensible special cases. When you understand that definition, everything comes into focus. Of course, one reason to choose that unifying definition over the original but simple definition might be that it is more useful part of our math practice. But still, that we have a unified definition might be very handy!
So, I take it that the deep answer to the student’s question will pick up either or both of usability theme or the unifying theme. But ymmv!
Let me add a simple sequence i remember
X^0= x^(1-1)
We know x^(a+b) is x^a times x^b so we can write above equation as
x^(1-1)= x^1 times x^(-1)
x^1 times x^(-1)= x ÷x
x ÷x = 1
Instead of thinking about X^n in isolation think about it multiplied by Y. X^2 Y is Y multiplied by X two times or XXY. X^1 Y is Y multiplied by X one time or XY. X^0 Y is Y multiplied by X no times or Y. What does X^0 need to be such that X^0 Y = Y?
Just like nothing added together is the number that doesn’t change other numbers when added to it, zero (0X = 0), nothing multiplied together is the number that doesn’t change things when multiplied, one.
How many ways can you organise 0 things? One.
Like stated
X^a * X^b = X^(a+b)
Try this for yourself with random values and see that this is always the case
X^a / X^b = X^(a-b)
Once again if you try this you will see it is always the case.
Now
X^a / X^a is X^(a-a)
answer: it makes sense for it to be the multiplicative identity when you consider x^n * x^m = x^(n+m)
Please show your teacher these explanations (in a very respectful way) – a 10th grade math teacher should have an answer for this question. Just don’t be a dick about it. Be like “hey I found an answer online for that math question, would you like to see it?”
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