I’m too dumb to get it
In: Mathematics
When you put a number to the power of something (2^2, 3^4, etc) you’re multiplying it that many times (2 x 2, 3 x 3 x 3 x 3, etc).
So, if you have a number to the power of 0, you’re multiplying that number by itself 0 times.
You can have a number to the power of 1, this would give you the number itself (2^1=2, etc).
So 2^2 would be 2 about 2 times (2×2) but 2^0 would be 2 about 0 times.
So if we have the number 2, 0 times, then we have don’t have a number, which means we have 0.
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.
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?
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
WE HAVE 13 axioms in number theory. Things that everyone agree are taken by true. They don’t need proof. There’s nothing magic about these axioms, other set could have been chosen. But it was decided to have those 13.
As an example, one of them is: x * 1 = x . This is accepted by a true not needed to be proved.
And you have de definition: x ^ y = x * x * x… * x a number of y times, right?
Now we have: x ^ (y+z) = x^y * (x^z). It can be proved by induction, but it makes sense by the above definition.
and x ^ (y-z) = x^y / x^z. It also can be proved by induction.
So x ^ (y-y) = x^y / x^y .
and you have
x^0 = 1.
For multiplication, 1 is the neutral number, just like 0 is the neutral number for addition. When you take the take the 0th power of a number, you bring it back down to neutral, and so that makes it 1. It’s not supposed to make sense, it’s just an arbitrary base case to make the rest of algebra work.
There were better explenations here but here is another angle. You can also look at it in terms of an empty product.
If you look at an empty sum, then it should be the neutral element for addition, so 0. (Add 0 to any number and it stays the same). So if add up a number 0 times to it self, it’s 0. Look at 0*x = 0 for any number x.
By the same logic an empty product should be the neutral element of multiplication, so 1. (Multiply any number with 1 and it stays the same). So if you multiply a number 0 times by it self, It’s 1. Look at x^0 = 1 for any number x.
*x^n* is the number of times you have to divide *x* out of *n* to get *1*.
125 = 5^3
25 = 5^2
5 = 5^1
1 = 5^0
1/5 = 5^(-1)
1/25 = 5^(-2)
Can I get some ELI2?
Latest Answers