It kinda doesn’t make sense.
x¹= x
x² = x*x
x³= x*x*x
etc…
and even with negative numbers you’re still multiplying the number by itself
like (x)-² = 1/x² = 1/(x*x)
In: 1797
Simplest answer.
Any number x can be rewritten as x*1.
So x^n = x * x *…x*1
Whether the number of x is equal to n. So write out x^0 in the same way.
x^0 = 1
Think of it as 0 = n – n
So x to the power of 0 is x to the power of (n -n)
So
X^0 = x^(n-n)
Which is x^n ÷ x^n
Thus by definition it is 1 unless x is 0.
since the other explanations perhaps didn’t take / didn’t address the fundamental concept/feeling/intuition:
consider the power operator to do the following:
2^3 = 1*2*2*2
as in 1 * (multiply 2 by itself 3 times)
2^0 = 1 (followed by no multiplications of 2)
1* (multiply 2 by itself 0 times) = 1.
What others have written is correct, i am just presenting a different approach which uses limits:
So, just open the calculator app and use the power function with any number if you want,
Let’s take 2 in our example,
so (2)² = 4,
(2)¹ = 2
Keep decreasing the power
(2)⁰•⁹⁵ = 1.9318
(2)⁰•⁵ = 1.4142
(2)⁰•¹ = 1.0717
And so on…
So you can observe, in the equation (2)ˣ = y
As x tends to 0 ( X —-> 0)
We can see y tends to 1 ( y —-> 1)
Hence when x = 0, y should be 1
X^3 is X * X * X
If i do X^3 / X^2
Thats just X^1, and also the same thing as me subtracting exponents.
So if i do X^3 / X^3 thats equal to 1, no argument there right?
But also when we do exponent division, we can just subtract the exponents.
So we’re left with X^0
x^y >> How many times (y) do you multiply x by itself.
Any integer x == x•1
If x equals 2 and y equals 3 we get (2•1)³
Taking the 1 out front 1•2•2•2 so we have 1 times the ’the answer to the question’ “what do you get when you multiple 2 by itself 3 times. The answer is 8 so >> 1•8
In the case of 2⁰ we have 1 times the answer to the question “what do you you get when you DON’T multiply 2 by itself ie what is “1 times nothing” not “1 times 2 times 0”
0 does not equal nothing in math. And i think this might be where your misunderstanding is coming from.
Read this for more clarity: http://www.differencebetween.net/language/words-language/difference-between-zero-and-nothing/#:~:text=%E2%80%9CZero%E2%80%9D%20is%20considered%20to%20be,that%20number%20in%20numerical%20values.
So 1 times nothing or an empty set is still 1.
If you want to think of it in terms of spoons or something it’s difficult.
But I’ll try my best.
If you have a spoon on your table and you want to make 7 rows of spoons you’ll have 7 spoons. If you have 1 spoon and make zero rows of spoons you’ll have to take your 1 away. And have zero left.
However if you have 1 row of 1 spoons and add 8 rows of 1 fork each you still have 1 spoon. Because multiplying it by a irrelevant set (an effectively empty set of spoons) doesn’t change the fact that you still only have one spoon.
I hope that helps.
I think that best way to see it, is to just halve the power and see where it gets you.
2^4 = 16
2^2 = 4
2^1 = 2
2^(1/2) = 1.414
2^(1/4) = 1.189
2^(1/8) = 1.091
.
.
.
2^(1/1000) = 1.001
The value is approaching 1
If you then flip it over to negative exponent (-1/1000, -1/8, -1/4 …) you will see it continues past 1 into smaller values. Making 0 exponent undefined would leave an undefined value in otherwise continuous function.
For the same reason x * 0 = 0 instead of non-existent. When you “repeat addition” 0 times (you can interpret multiplication as repeated addition), you get the additive identity, which is 0. Repeated addition looks like: 0 + x + x + x + …. When you do x * 3 you take the first four elements of that series, 0 + x + x + x. When you do x * 0, you take the first element of that series, 0.
The same holds with x^0 . When you “repeat multiplication” 0 times (you can interpret exponentiation as repeated multiplication), you get the multiplicative identity, which is 1. Repeated multiplication looks like: 1 * x * x * x * …. Doing x^3 takes the first four elements of that series, and doing x^0 takes the first element of that series.
You can easily understand this in non-mathematical terms:
– Imagine you have a baby and a dog and every time the dog barks, the baby cries 3 times. If the dog barks twice, how many times has the baby cries? 6. That’s because 3 * 2 = 6. If the dog barks zero times, how many times has the baby cried? Zero. That’s because 3 * 0 = 0.
– Similarly, imagine you have a coupon that lets you take 50% off the price, and they don’t limit you to one coupon. How much of the full price do you pay with two coupons? 25%. That’s because 0.5^2 = 0.25. How much of the full price do you pay if you have zero coupons? 100%. That’s because 0.5^0 = 1. Your store would quickly go out of business if you said, “hey, since you have zero coupons, the price is non-existent, so you can’t buy it.”
In addition to other comments, I’d like to add that powers of 10 are helpful for remembering this for me — since the rule is, for powers of 10, the exponent tells you how many 0’s are in the number. Positive exponents, the 0’s come after the 1; negative, they come before. But if the exponent itself is 0, well, there are no 0’s, before or after the 1. It’s just a 1 🙂
Disclaimer: I don’t think that this will elif, but it may still help.
*x = x^1 = x^0+1 = x^0 * x^1 = x^0 * x*.
So we have that
*x = x^0 * x*.
Divide both sides by *x* to get that
*1 = x^0 .*
I should note that we assume that *x* is not zero, but I don’t think that’s important for this explanation
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