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
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I was on the hiring committee for the math department of a community college and we would role play students in a lesson about exponential functions given by the applicant. We would ask your question. An answer no better than what your teacher gave was close to eliminating any hiring chance by itself. Reasonable answers include some aspects of the following:
It is a definition and could be defined however one wants. Defining it as 1 is the most useful definition mathematically for both consistency and usability. For example,
since anything other than zero divided by itself is 1 we have x^n / x^n is 1. But we also have a rule that it should equal x^(n-n) = x^0. For that rule to still work we need x^0 =1. It also maintains the pattern of making the exponent one smaller mean that the result is divided by x. e.g. going from x^3 to x^2 one divides by x. Going from x^2 to x one divides by x. Going from x^1 to x^0 one divides by x and x/x = 1. A similar idea leads to why x^(-n) = 1/x^n which is allows scientific notation to handle very small as well as very large numbers. One more thing to consider is an analogy with addition. Adding 0 to a number does not change a number (does nothing). Exponentiation is about multiplication and for multiplication multiplying by 1 does not change a number (does nothing) so you can think of a zero exponent as doing nothing in terms of multiplication.
Think of it like this: x^3 is a cube with a side of length x and volume x^3. X^2 is a square with a side of length x and area of x^2. X^1 is a line of length x. X^0 is a point, one point.
When you look at the exponent as multiplication, also multiply by 1 as well, and it may help visualize why.
x³ = x * x * x * 1
x² = x * x * 1
x¹ = x * 1
x⁰ = 1
X^a * X^b = X^(a+b)
Therefore, X^0 * X^n = X^(0+n) = X^n.
What is the number which, multiplied by X^n, results in X^n ? 1. Therefore, X^0 = 1.
1 is the multiplicative identity, as in x * 1 = x. Because exponentiation is repeated multiplication, you start with this identity before multiplying your first x.
Similarly 0 is the additive identity, because x + 0 = x. Multiplication is repeated addition, so you start with 0 before adding your first x. This is another reason why x * 0 = 0.
Fucking woof at that response from your teacher, damn. For the record, x to the power of y is not x * y, but (x * x) y number of times.
For example, 2^4 is not (2 * 4), but (2 * 2 * 2 * 2)
You put 1000 bucks in a bank with an annual interest rate of 5%.
How much do you have after *n* years?
After 1 year, it’s 1000*1,05=1050.
After two years, it’s 1000*1,05*1,05=1102,5.
And so on.
After *n* years it’s 1000*1,05*^(n)*. So the 1,05*^(n)* expresses how much your capital has multiplied after *n* years.
What’s 1,05^(0) then? It’s how much your capital multiplied after 0 years, i.e. you put it in and immediately withdrawn. Obviously, it’s the same amount, so it’s multiplied by 1. That’s why 1,05^(0)=1.
It works the same way regarldess of what the annual interest rate is, so that’s why x^(0)=1 for any x.
“Raised to a power” kind of means “how many times is the base number multiplied by itself?” So, you get something like 3² = 3×3 = 9, or 3³ = 3x3x3 = 27, right? The power expresses “how many copies of the number are multipled together.”
When you multiply 0 copies of the base number, you’re not left with the additive identity, 0, but the multiplicative identity, 1.
3³ = 1x3x3x3, and 3² = 1x3x3, and 3¹ = 1×3 and 3⁰ = 1 with no copies of 3 to multiply by.
So, for your question, you aren’t multiplying by zero, which multiplicatively turns the equation to zero, you’re adding zero copies of the base number into a multiplicative expression, where the multiplicative identity, 1, can always exist without changing the result. Since 1 is the only thing in the expression (as you didn’t add any copies of the base number to the expression), the result is just 1.
it’s actually rlly cool lol. so yk how if you have like 2 to the 5th divided by 2 to the 3rd.. 5-3=2 so you get 2 to the 2nd. now say you have 2 to the 3rd divided by 2 to the 3rd. 3-3=0 so you get 2 to the 0. but you ALSO just divided a number by itself, which is ALWAYS 1. this applies to any number too
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