I’m doing my GED and passed that math part but this never came up. I saw it on a meme and am lost.
Help!!??
In: 9770
There are some simple proofs of this fact, and here is one:
x^(n)/x^(n) = 1 (this is just a simple fact; any number over itself equals 1)
If you do that division following exponent rules, you get:
x^(n-n) = 1
x^(0) = 1
This is a general rule, so it can be applied to other numbers too. Anything the power of 0 is 1.
There’s a few ways to think about exponents. Method 1 is to see 2^0 as “2 times itself 0 times”. In my mind I see this as “2 divided by itself” which gives one.
since we know that
> x^n / x^m = x^(n-m)
we can say that
> 2^1 = 2
> 2^1 / 2^1 = 2^(1-1) = 2^0
> 2 / 2 = 1
therefore 2^0 = 1
This has been asked a bunch before so I’m repeating this explanation from u/AxolotlsAreDangerous
x^5 / x^3 = (x* x* x* x* x)/(x* x* x)
=x^2
In general, x^n /x^m = xn-m
Let m=n, this property should still hold.
x^n /x^n =x^n-n =x^0
Any number divided by itself is 1, x0 =1.
1 is, in a sense, to multiplication what 0 is to addition. They’re the identity element , meaning x* 1=x and x+0=x.
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2 ^ 4 = 16
2 ^ 3 = 8
2 ^ 2 = 4
2 ^ 1 = 2
16 -> 8 -> 4 -> 2
Each number in the sequence is the previous one divided by 2. So we can continue the sequence:
2 ^ 0 = (2 / 2) = 1
Here.
If 2^4 = 2x2x2x2 =16.
Then 2^3 = 16/2 = 2x2x2x2/2 = 2x2x2 = 8.
And 2^2 = 8/2 = 2x2x2/2 = 2×2 = 4.
So 2^1 = 4/2 = 2×2/2 = 2
So 2^0 = 2/2 = 1.
And that’s why 2^-1 = 1/2
And 2^-2 = 1/(2×2) = 1/4.
Etc
Powers are repeated multiplication.
So 2^1 = 2
2^2 = 2×2 = 4
2^3 = 2x2x2 = 8
2^4 = 2x2x2x2 = 16
If you look at the list above, but go backwards, you see that when I reduce the power, I multiply by 2 one fewer time. Equivalently, I divide by 2, or halve the number.
2^0 would be the next term in this sequence, and would be half of 2^1.
2^1=2, so 2^0 is half of that, 1.
We can go even further into negative powers, which is now explicitly dividing.
So 2^-1 = 1/2
and 2^-2 = 1/4.
—–
If you are worried about how ‘repeated multiplcation’, starts at 1, note that multiplcation by 1 makes no difference, so it is therefore the starting point for any multiplcation.
That is, I started earlier with 2^1 = 2.
We can multiply by 1, and this makes no difference, so
2^1 = 1×2
So by ‘repeated multiplcation’, we really mean ‘repeated multiplication *starting at 1*’, that is why 2^0=1, since we start with 1 and then do nothing to it.
Another helpful way to think about it:
2^2 = 2 * 2 * 1
2^1 = 2 * 1
2 ^ 0 = 1
2^(-1) = 1/2
2^(-2) = 1 / 2 / 2 = (1/2)^2 = (1^2 )/(2^2 ) = 1/2^2
Consider the division of powers: 2^3 / 2^3.
If we use the divide rule, we subtract exponents so it would be 2^(3-3) = 2^0
If we just find the values (2^3 = 8), then it’s 8/8 which is 1.
If one quotient gives two answers, those answers must be the same so therefore, 2^0 = 1.
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