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
Hiya. I think this question usually doesn’t get answered with a fully true answer. Hopefully this explanation makes some sense.
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When you raise a number X to a power Y, it means you multiply the number X by itself Y times, right? Like 5^3 means 5×5×5, we multiply 5 by itself 3 times.
So what is 5^0? Can we multiply 5 by itself zero times? That’s a nonsense statement, it doesn’t really mean anything.
So maybe we can clarify a bit: when Y is a natural number (1,2,3,4,…), raising X to the power of Y means we multiply X by itself Y times.
Now, we still want to get some kind of answer when we try 5^0, or even 5^-2, even though multiplying 5 by itself 0 or -2 times is nonsense. So, we see if there’s a pattern, and then we adjust the definition of raising to a power to fit that pattern.
The pattern we might notice is this:
If we go from 5^3 = 5×5×5 to 5^2 = 5×5, we divided by 5. If we go from 5^2 = 5×5 to 5^1 = 5, we again divided by 5.
So what happens if we continue the sequence down by just following this pattern and dividing by 5 over and over?
5^3 = 5×5×5 = 125
5^2 = 5×5 = 25 ( = 5^3 ÷ 5)
5^1 = 5 = 5 ( = 5^2 ÷ 5)
5^0 = ?? = 1 ( = 5^1 ÷ 5)
5^-1 = ?? = 0.2 ( = 5^0 ÷ 5)
5^-2 = ?? = 0.04 ( = 5^-1 ÷ 5)
While the original definition “you multiply 5 by itself that many times” stops making sense, the pattern of “you divide by 5 to get to the next one” can just keep giving us the next numbers in the sequence.
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At the end of the day, the answer to “why does 5^0 = 1” is “while it’s impossible to multiply 5 by itself zero times, if we just say it’s true that 5^0 = 1, it fits nicely with our pattern. So, we say it’s true”
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