It kinda doesn’t make sense.
x¹= x
x² = x*x
x³= x*x*x
etc…
and even with negative numbers you’re still multiplying the number by itself
like (x)-² = 1/x² = 1/(x*x)
In: 1797
Eli5: Rule of dividing exponents. X^m / X^n = X^m-n.
So X^a / X^a = X^a-a = X^0
Any number divided by itself equals 1, therefore X^a / X^a = 1
Transitive X^0 = 1
Since it’s ELI5, the quickest way to explain it is that every number going up is *x, and every number going down is /x, meaning x^0 = x^1 /x, which results in x/x=1
i think of it as, for example
4^1 = 4 and 4^-1 = 1/4
because of laws of indices, multiplying 4^1 by 4^-1 gives you 4^0 (you add the powers)
as 4 multiplied by 1/4 = 1
Oh my god. I am so glad you asked this question. I did not understand it either up until now, thank you Sir
I’ll try to explain it how my high school math teacher explained it to us.
x^(0)
=> x^(y-y) ( This just means that we can write 0 as a subtraction of 2 same numbers; y in this case)
=> x^(y) / x^(y) ( A rule of exponents says that if you have different exponents to the same number in division, the exponents can be subtracted and vice versa)
=> 1
OK so, when powers raise by 1, e.g. From x^2 to x^3, it multiplies by x. Therefore, when powers decrease by 1, it divides by x. And therefore, x^1 divided by x equals 1. So x^0 is 1.
Revise the law of indices which states,
x^n / x^m = x^(n – m)
So,
x^0 = x^(1 – 1) = x^1 / x^1 which cancels to 1.
Hence proved
Another way to think of this:
x^a * x^b = x^(a+b)
This is true for any positive values of a and b, so think about what x^0 needs to be to maintain this relationship.
if x^a * x^b = x^(a+b) , and b is zero, then you need
x^a * x^0 = x^(a+0) = x^a , which means x^0 must be 1.
You can then extrapolate that relationship with negative numbers as well.
Example, 2^5 is 32
2^5 * 2^-3 = 2 ^(5-3) = 2^2 = 4
So for this property to be consistent not just with positive exponents, but also negative exponents, this is the formulation we use.
To be clear, the notation of exponents is created by humans, but we want to create mathematical rules that are logical, consistent, and when feasible a useful description of reality.
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