Easiest explanation I can think of using the division law for exponents:
Since we can use any number for the initial fraction, as long as the denominator is the same as the numerator, any number to the zeroth power is equal to 1. In general terms, then, for any number, x:
You can see this in the example above but perhaps it’s better to use different powers to make things a bit clearer.
2^5=2x2x2x2x2
2^3=2x2x2
(25)/(23)=(2x2x2x2x2)/(2x2x2)
You can cancel 3 of the 2s from the top and bottom of the fraction to be left with 2x2, or 2^2.
I.e. (25)/(23)=2^2
The quicker way to calculate this is doing 2^(5-3) which when you resolve the bracket is obviously just 2^2 or 2x2.
If both numbers in the bracket are the same the bracket will always resolve to 0, which is the same as saying a number divided by itself, any number divided by itself is one so it follows that any number to the power 0 is also 1 (because it’s essentially exactly the same calculation).
0 is the neutral element for addition. This is why when we have a number then 0 + number = number (0 doesnt change the value in addition) and why 0 x number = 0 (if you add a number 0 times you will have 0). (Multiplication is adding one of the numbers to itself the number of times designated by the second number)
The same way 1 is the neutral element for multiplication. This is why when you have some number then 1 * number = number. This is also why number^0 = 1 (if you never multiply by a number you are left with the neutral element. It would be weird if powering by 0 left you with 0 for example because of how negative powers work)
This is the level 1 answer.
The level 0 answer is that it is this way because all of mathematics is a construct designed to ease problem solving and all people collectively agreed that doing it this way is way more useful (because it is)
You can think of 1 as the “empty product” (or the “neutral element of multiplication” if you want to be fancy). 2^x means you have x factors of 2. If you have 0 factors, you have the “empty product”
I like to think of it this way:
2^3 is the same as 2 x 2 x 2.
But you can arbitrarily multiply by as many 1s as you want because 1 has the identity property for multiplication.
So we can also write 2^3 as 2 x 2 x 2 x 1 x 1.
2^2 as 2 x 2 x 1 x 1.
2^1 as 2 x 1 x 1.
2^0 as 1 x 1 or just 1.
Multiplying a number by another number is the same as adding a number to itself that many times. And 0 is has the identity property for addition, so similarly:
2 x 3 = 2 + 2 + 2 + 0 + 0
2 x 2 = 2 + 2 + 0 + 0
2 x 1 = 2 + 0 + 0
2 x 0 = 0 + 0
The simplest way I think of it is by the properties of exponentials:
2^3 / 2^2 = (2 * 2 * 2) / (2 * 2) = 2 = 2^(3-2)
Dividing two exponentials with the same base (in this case 2) is the same as that same base (2) to the power of the difference between the exponent in the numerator minus the exponent in the denominator (3 and 2 in this case).
Well looks like some people already answered your question but let me show you quick proof video that may help you understand how powers work: https://youtu.be/kPTp82EGjv8?feature=shared
I know I’m bad at math but I don’t understand how 2x0=0 but 2^0=1
How are they different answers when they’re both essentially multiplying 2 by zero?
Someone with a bigger brain please explain this
Edit: I greatly appreciate all the explanations but all they’ve done is solidify the fact that I’ll never be good at math 😭
Easiest explanation I can think of using the division law for exponents:
Since we can use any number for the initial fraction, as long as the denominator is the same as the numerator, any number to the zeroth power is equal to 1. In general terms, then, for any number, x:
This isn’t strictly speaking a proof, but it did help me to accept it as it demonstrates the function that makes it 1.
2^3 = 2x2x2
2^2 = 2x2
(23)/(22) = (2x2x2)/(2x2) = 2
= 2^(3-2)
In general terms:
(xa)/(xb) = x^(a-b)
If a and b are the same number this is x^0 and obviously (xa)/(xa) is one because anything divided by itself is 1.
Hope that helps
Yes, of course, obviously…JFC, what??
deleted by creator
2^(a-b) = (2a)/(2b)
You can see this in the example above but perhaps it’s better to use different powers to make things a bit clearer.
2^5=2x2x2x2x2
2^3=2x2x2
(25)/(23)=(2x2x2x2x2)/(2x2x2)
You can cancel 3 of the 2s from the top and bottom of the fraction to be left with 2x2, or 2^2.
I.e. (25)/(23)=2^2
The quicker way to calculate this is doing 2^(5-3) which when you resolve the bracket is obviously just 2^2 or 2x2.
If both numbers in the bracket are the same the bracket will always resolve to 0, which is the same as saying a number divided by itself, any number divided by itself is one so it follows that any number to the power 0 is also 1 (because it’s essentially exactly the same calculation).
Rule = #^0 = # x 1
Don’t ask why…got it.
No not quite, #^0 = 1
Wait, so 5^0 = 1??
Yup
5^0 can be rewritten as 5^(2-2)
5^(2-2) = (52)/(52)
This is a number divided by itself so cancels to 1 every time, regardless of #.
0 is the neutral element for addition. This is why when we have a number then 0 + number = number (0 doesnt change the value in addition) and why 0 x number = 0 (if you add a number 0 times you will have 0). (Multiplication is adding one of the numbers to itself the number of times designated by the second number)
The same way 1 is the neutral element for multiplication. This is why when you have some number then 1 * number = number. This is also why number^0 = 1 (if you never multiply by a number you are left with the neutral element. It would be weird if powering by 0 left you with 0 for example because of how negative powers work)
This is the level 1 answer.
The level 0 answer is that it is this way because all of mathematics is a construct designed to ease problem solving and all people collectively agreed that doing it this way is way more useful (because it is)
Choose which one you want
Fuck me this is the only one I understand 😭
You can think of 1 as the “empty product” (or the “neutral element of multiplication” if you want to be fancy). 2^x means you have x factors of 2. If you have 0 factors, you have the “empty product”
I like to think of it this way:
2^3 is the same as 2 x 2 x 2.
But you can arbitrarily multiply by as many 1s as you want because 1 has the identity property for multiplication.
So we can also write 2^3 as 2 x 2 x 2 x 1 x 1.
2^2 as 2 x 2 x 1 x 1.
2^1 as 2 x 1 x 1.
2^0 as 1 x 1 or just 1.
Multiplying a number by another number is the same as adding a number to itself that many times. And 0 is has the identity property for addition, so similarly:
2 x 3 = 2 + 2 + 2 + 0 + 0
2 x 2 = 2 + 2 + 0 + 0
2 x 1 = 2 + 0 + 0
2 x 0 = 0 + 0
In addition to the explanation others have mentioned, here it is in graph form. See the where the graph of 2^x intersects the y axis (when x=0):
https://people.richland.edu/james/lecture/m116/logs/exponential.html
This also has some additional verbal explanations:
http://scienceline.ucsb.edu/getkey.php?key=2626
The simplest way I think of it is by the properties of exponentials:
2^3 / 2^2 = (2 * 2 * 2) / (2 * 2) = 2 = 2^(3-2)
Dividing two exponentials with the same base (in this case 2) is the same as that same base (2) to the power of the difference between the exponent in the numerator minus the exponent in the denominator (3 and 2 in this case).
Now lets make both exponents the same:
2^3 / 2^3 = 8/8 = 1
2^3 / 2^3 = 2^(3-3) = 2^0 = 1
Well looks like some people already answered your question but let me show you quick proof video that may help you understand how powers work: https://youtu.be/kPTp82EGjv8?feature=shared