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 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. Comparing exponent expressions. Practice: Powers of fractions.
Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript - If we think about something like two to the third power, we could view this as taking three twos and multiplying them together, so two times two times two, or equivalently, we could say this is the same thing as taking a one and then multiplying it by two three times, so actually, let's just go with this definition right over here, and this, of course, is going to be equal to eight.
Now what would, based on this definition I just did, what would two to the second power be? Well, this would be one times two twice, so one times two times two, which, of course, would be equal to four. What would two to the first power be? Well, that would be one, and we would multiply it by one two, one times two, which, of course, is equal to two. Now let's ask ourself an interesting question. Based on this definition of what an exponent is, what would two to the zeroth power be?
I encourage you to just think about that a little bit. If you were the mathematics community, how would you define two to the zero power so it is consistent with everything that we just saw.
Well, the way we just talked about it, we just said exponentiation is you start with a one and you multiply it by the base zero times, so we're not gonna multiply it by any two, so we're just gonna be left with a one. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question:.
Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'.
For example, 2 3 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1. So what does 3 0 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table?
Just one I know this sounds a little fishy since we started with a rule I could have just made up which is why I gave the other reason first , but these formulas are all consistent and there is never any magic step, I promise! One rule for exponents is that exponents add when you have the same base. Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:.
If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:. Let's look at what it means to raise a number to a certain power: it means to multiply that number by itself a certain number of times. Let's look at a few examples:.
If you look at the pattern, you can see that each time we reduce the power by 1 we divide the value by 3. Using this pattern we can not only find the value of 3 0 , we can find the value of 3 raised to a negative power! Here are some examples:. No matter what number we use when it is raised to the zero power it will always be 1. Suppose instead of 3 we used some number N, where N could even be a decimal.
0コメント