Question

when is x°=1 and how can we prove it​

Answer 1

This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about what a power means. When we raise a number to the nth power, that really means that we multiply that number by itself n times, so for example, 22 = 2*2 = 4, 23 = 2*2*2 = 8, 34 = 3*3*3*3 = 81, and so on. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all! What does that mean? Well, let's go even farther back to the simplest case: addition. What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same. In short, 0 is the only number such that for any number x, x + 0 = x. So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number. In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.