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Course: 8th grade>Unit 1

Lesson 5: Exponents with negative bases

The 0 & 1st power

Different ways of thinking about exponents. Raising a number to an exponent means multiplying that number by itself a certain number of times. Any non-zero number raised to the zero power will be equal to one, and that any number raised to the first power will be equal to itself. Created by Sal Khan.

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• how do you type the multiplication symbol on a keyboard?
(131 votes)
• Well, there are two ways that I know of:
- An asterix = * (which, by the way, is from a Greek root word that means "little star") You can usually find it where your numbers are on your keyboard. Ex: 12 * 6 = 72
- A lower case X = x. I don't think I have to tell you where that is on a keyboard! Ex: 12 x 8 = 96
Also as a footnote to this, "x" will sometimes not work because it could also represent a variable. So be careful!
Hope this helps! :)
(314 votes)
• Why any number (except 0 ) raise to power 0 is always 1 ?
(16 votes)
• It has to do with the properties of exponents.
Here's a good example:
Lets say you have a number a^x/a^y and x = y.
Using a property of exponents you can rewrite the equation as a^(x-y) and since x=y that becomes a^0.
Also since x = y, a^x = a^y and so it becomes 1/1 or just 1.
(5 votes)
• At Sal says it will make sense. I never really got past the 2 to the 0 power. I don't understand how he took two, put a zero above it, and then turned it into a one?! How does that work?
(13 votes)
• What he showed helped me understand why it does that, so now let me try to explain it for you:
He changed the way he did the exponents to multiplying `1 times` how many numbers (the number that the exponent is) to that one.

So when it is `2⁰ = 1` because there aren't any `2`s to multiply by.
So something as big as `1,000,000⁰ = 1`

Now let's do it regular:
When you have `4⁶ = 1 × 4 × 4 × 4 × 4 × 4 × 4 = 4096`
Well, that is a bit too big of a number so let's do `3² = 1 × 3 × 3 = 9`

Ask me to clarify anything.
(24 votes)
• What does the little circle above the 2 mean?
(5 votes)
• The circle above is the exponent. The rule is that any number raised to the power of 0 equals to 1. So if 2 or 1,000,000 is raised to the power of 0 it equals 1.
(16 votes)
• Why incorporate the 1 at all?
(10 votes)
• Because incorporating the 1 helps to give more intuition for why every number raised to the power of 0 equals 1. Sure, you can divide 2 by 2 to get 1, however, you can also say "hey, I have no 2's, but still have the 1."
(3 votes)
• math isnt mathing anymore.
(11 votes)
• bruh I understood it for a hot second than you hade to do THIS
(8 votes)
• Please help. I don't understand how you get 1 for the answer wouldn't be 2 since you aren't multiplying it by itself?
(6 votes)
• Not quite, you see we multiply an exponent by whatever number is in that position ,so if zero is in the position The number multiplies itself zero times ,And that number is just zero . But since there is an invisible one beside the number ,That makes the answer one instead of zero.
(3 votes)
• () I really don't understand how any non-zero number to the power of 0 is 1.
(8 votes)
• because your not multiplying it by itself any times
(0 votes)
• I understood it but now I don’t
(8 votes)
• It might take a while for it to fully 'sink in' to your brain. Perhaps try reviewing the video a couple of times to see if it gets any clearer.
Hope this helps!
-Catlovertheaspiringvet🩺🐈
(0 votes)

Video transcript

What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples. So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them. So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8. So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1. So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times. So this is times negative 2 times negative 2 times negative 2. So clearly these are the same number. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8. And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power. So let's think about that a little bit. What is positive 2 to the-- based on this definition-- to the 0 power going to be equal to? Well, we just said. This says how many times are going to multiply 1 times this number? So this literally says, I'm going to take a 1, and I'm going to multiply by 2 zero times. Well, if I want to multiply it by 2 zero times, that means I'm just left with the 1. So 2 to the zero power is going to be equal to 1. And, actually, any non-zero number to the 0 power is 1 by that same rationale. And I'll make another video that will also give a little bit more intuition on there. That might seem very counterintuitive, but it's based on one way of thinking about it is thinking of an exponent as this. And this will also make sense if we start thinking of what 2 to the first power is. So let's go to this definition we just gave of the exponent. We always start with a 1, and we multiply it by the 2 one time. So 2 is going to be 1-- we're only going to multiply it by the 2. I'll use this for multiplication. I'll use the dot. We're only going to multiply it by 2 one time. So 1 times 2, well, that's clearly just going to be equal to 2. And any number to the first power is just going to be equal to that number. And then we can go from there, and you will, of course, see the pattern. If we say what 2 squared is, well, based on this definition, we start with a 1, and we multiply it by 2 two times. So times 2 times 2 is going to be equal to 4. And we've seen this before. You go to 2 to the third, you start with the 1, and then multiply it by 2 three times. So times 2 times 2 times 2. This is going to give us positive 8. And you probably see a pattern here. Every time we multiply by 2-- or every time, I should say, we raise 2 to one more power, we are multiplying by 2. Notice this, to go from 2 to the 0 to 2 to the 1, we multiplied by 2. I'll use a little x for the multiplication symbol now, a little cross. And then to go from 2 to the first power to 2 to the second power, we multiply by 2 and multiply by 2 again. And that makes complete sense because this is literally telling us how many times are we going to take this number and-- how many times are we going take 1 and multiply it by this number? And so when you go from 2 to the second power to 2 to the third, you're multiplying by 2 one more time. And this is another intuition of why something to the 0 power is equal to 1. If you were to go backwards, if, say, we didn't know what 2 to the 0 power is and we were just trying to figure out what would make sense, well, when we go from 2 to the third power to 2 to the second, we'd be dividing by 2. We're going from 9 to 4. Then we'd divide by 2 again to go from 2 to the second to 2 to the first. And then it seems like we should just divide by 2 again from going from 2 to the first to 2 to the 0. And that would give us 1.