why is everything to the power of 0 1?
In: 3
When you add together no elements, you end up with 0, right? Well, what’s special about 0 with respect to addition? The answer is that 0 is the “do nothing” number with respect to addition, 0 + x = x for any number x. Another way to see this is if you want to come up with a procedure to add together a list of numbers. You start by setting your counter to 0 and then go through the list adding each number to the counter. If the list is empty then you just end up with 0.
Now the question of x^(0). We’re essentially asking “what happens if you take 0 copies of x, put them in a list, and multiply them together?” Well the procedure for multiplying a list of numbers is that you start with the “do nothing” number – which in the case of multiplication is 1, since 1*x = x for any number x. Then you multiply that number by each number in the list. If your list happens to be empty then you just get 1, which is why x^(0) = 1.
(This is true even for x=0, which is one reason why in most contexts where mathematicians choose to define 0^(0) then they define it to be 1.)
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