I’m confused, shouldn’t it be 0?
In: Mathematics
Let’s take x^0 and rewrite it to make it easier to understand. If we multiply x^1 and x^-1 together, we add the exponents together: x^1 * x^-1 = x^(1-1) = x^0 .
x^1 = x and x^-1 = 1/x, so x^1 * x^-1 = x * 1/x = x/x = 1
Because x^0 = 1 makes a lot of rules work better.
For example set n=0 for the rule: x^n * x = x^n+1
Or set n = m for the rule: x^m / x^n = x^m-n
Let’s use 2^(n) as example. We know that 2^(1)=2, 2^(2)=4, 2^(3)=8, etc… In general, when you add 1 to the exponent, you multiply the overall number by 2. Now work backwards: if you subtract 1 from the exponent, you’d be dividing the number by 2. So from that logic, 2^(2) = 2^(3)/2 = 8/2 = 4. Continuing, we get 2^(1) = 4/2 = 2. And doing it again, 2^(0) = 2/2 = 1. You can do this with any number (except 0) to show that x^(0) = 1.
This also explains how negative exponents work. By continuing subtracting the exponent, we get 2^(-1)=1/2, 2^(-2)=1/4, and in general, 2^(-n)=1/2^(n).
> I’m confused, shouldn’t it be 0?
According to other exponent rules, no.
Consider x^n / x^n = 1. That is the same as doing x^n * x^(-n) = x^(n – n) = x^0.
The best way to see this is to use the addition and subtraction rules for multiplying and dividing exponents. Say you have x³/x² to do this you would use 3-2=1 and get x¹ which is just x. Now do the same but use x³/x³ we can see that this is the same numerator and denominator so we should get 1. Likewise if we use the rule from above we get 3-3=0. Thus x⁰=1.
Suppose you want to solve x^y=z to find y. You can think of that as asking how many times you have to multiply 1 by x to get z.
So for instance with 2^y =8. You have to multiply 1 by 2 three times to get eight: 1x2x2x2 = 8.
So how many times do you have to multiply 1 by anything to get 1? The answer is zero times – it’s already 1.
You basically follow the pattern:
>x^(n+1) = x * x^n
or the other way around
>x^(n) = x^(x+1)/ x
Now if you plug in the examples
>x^1 = x * x^0
as x^1 = x therefore x^0 must be 1. Or even more intuitively for the other example:
>x^0 = x^1/x
well x^1 = x and x/x = 1 so x^0 is equal to 1
I’ll illustrate it with an example:
Suppose we have a population of some species that reproduces asexually. Let’s say that each individual produces an offspring every hour.
1 hour from now, we will have twice the current population.
2 hours from now, we will have four (or 2^(2)) times the current population.
3 hours from now, we will have eight (or 2^(3)) times the current population.
Now, if I asked you what will the population size be in 0 hours, by what factor would you multiply?
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