I’m too dumb to get it
In: Mathematics
Don’t think of exponents as simply being a number multiplied n times itself (If n=3, then 2x2x2). Think of exponents using one extra number: 1 (allowed under Identity Property of multiplication – not important here). So 2 squared is 1x2x2=4. 2 cubed is 1x2x2x2=8. So if n=0, then 2ⁿ is 1x(zero 2s) which equals 1. Keep in mind that zero 2s means it doesn’t exist, not 2×0. So any number with an exponent of 0 is 1 times nothing else. 1.
x^n means “Multiply 1 by x n times”
Math|Math Expanded|Words
:–|:–|:–
2^4|1x2x2x2x2 = 16|Multiply 1 by 2 four times
2^0|1|Multiply 1 by 2 zero times
How many times can you count a number 0 times? No times. But that set of nothing is still has nothing. Which is counter as 1. You count nothing
new problem:
why is 0! == 1?
How bout 0⁰?
0 to the power of anything is 0 but anything to the power of 0 is 1.
x^0 is something called an [empty product](https://en.m.wikipedia.org/wiki/Empty_product), that is, the result of multiplying zero numbers together. It is defined to be 1, the multiplicative identity (x • 1 = x), because that is the most useful and intuitive way of defining it. In any multiplication problem, let’s say (2 • 3), you can say there’s an implicit (2 • 3 • 1 • 1 • 1 • 1. . .) at the end. If you multiply zero number together, you’re left with only the (• 1 • 1 • 1 • 1. . .), so you get one.
So 2 to the power of 3 is 2x2x2 so 2 times itself 3 times
2 to the power of 1 would = 2
2 to the power of zero isn’t zero but it’s also not 2
If you did 2 times 2 times 2…… 0 times you just get 1
This is my understating of it
4^3 = 4x4x4 =64
4^2 = 4×4 = 16
4^1 = 4
4^0 = 4/4 = 1
4^-1 = 4/4/4 = 1/4
I love that this question was posted this week, because I just had to give my 5th grade students the answer of “I don’t know why 10 to the zero power is 1, as I would assume it would equal zero; let’s just say it’s one of those things in math that ‘just is’ until you learn more advanced concepts next year and beyond.” I honestly did do some research and found several of the explanations offered in the comments here. Unfortunately, they would be beyond the scope of where our curriculum is focused (they have just been introduced to the concept of exponents), and the abilities of the vast majority of my students.
My question is, is it conceptually disingenuous (or flat-out incorrect) to teach 11 year-olds who are just learning about multiplying and dividing by powers of ten that, for example, 10 squared = 10 x 10, or that two 10s are being multiplied together, and leaving it at that?
Exponents are a convenient way of showing repetitive multiplication of the same number. This is similar to multiplication showing repetitive addition of the same number.
so 3^(4) means I am going to multiply the base 3 by itself 4 (the exponent) times.
Whenever you multiply, 1 is a factor. A factor is an integer that can be multiplied by another integer to result in the original number.
So 10 = 2 x 5. but also 10 =1 x 2 x 5; 10=1 x 10
Now going back to exponents:
3^(4) = 3 x 3 x 3 x 3 By definition of exponential notation.
but we also need to remember that 3^(4) = 1 x 3 x 3 x 3 x 3. I like to call 1 the “invisible number of multiplication). It’s important to remember it is there.
so let’s look at a progression of exponents:
3^(4) = 1 x 3 x 3 x 3 x 3.
3^(3) = 1 x 3 x 3 x 3
3^(2) = 1 x 3 x 3
3^(1)= 1 x 3
3^(0) = 1
3^(0) indicates a multiplication operation where 3 is multiplied 0 times. But since it is multiplication, there must be some factor. Or yeah, 1 is always a factor in multiplication!
Also by the way:
3^(1)= 1 x 3 = 3 Any number to the first power = itself. Also the identity property of 1 says that any number times 1 =
itself.
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