Patterns in raising 1 and -1 to different powers | Pre-Algebra | Khan Academy

Let’s think about exponents with ones and zeroes. So let’s take the number 1 and let’s raise it to the eighth power.

So we’ve already seen that there’s two ways of thinking about this.

You could literally view this as taking eight 1’s and then multiplying them together.

So let’s do that.

So you have one two, three, four five, six, seven, eight 1’s and then you’re going to multiply them together And if you were to do that, you would get well 1 times.

1 is 1 times 1.

It doesn’t matter how many times, you multiply 1 by 1.

You are going to just get 1, You are just going to get 1 And you could imagine I did it eight times.

I multiplied eight 1’s, But even if this was 80 or if this was 800 or if this was 8 million.

If I just multiplied 1, if I had 8 million 1’s and I multiplied them all together, it would still be equal to 1. So 1 to any power is just going to be equal to 1 And you might say, hey what about 1 to the 0 power.

Well, we’ve already said anything to 0 power, except for 0 that’s, where we’re going to it.’s actually up for debate, But anything to the 0 power is going to be equal to 1 And just as a little bit of intuition.

Here you could literally view this as our other definition of exponentiation, which is you start with a 1, and this number says how many times you’re going to multiply that 1 times.

This number So 1 times 1 zero times is just going to be 1, And that was a little bit clearer when we did it like this, where we said 2 to the let’s say, fourth, power is equal to this was the other definition of exponentiation.

We had, which is you start with a 1, and then you multiply it by 2, four times so times, 2 times 2 times 2 times 2, which is equal to let’s see.

This is equal to 16.

So here, if you start with a 1 and then you multiply it by 1 zero times, you’re still going to have that 1 right over there And that’s.

Why anything that’s not 0 to the 1 power is going to be equal to 1? Now let’s try some other interesting scenarios.

Let’s start try some negative numbers.

So let’s take negative 1 And let’s first raise it to the 0 power. So once again, this is just going based on this definition.

This is starting with a 1 and then multiplying it by this number 0 times.

Well, that means we’re just not going to multiply it by this number.

So you’re just going to get a 1.

Let’s try negative 1.

Let’s try negative 1 to the first power.

Well, anything to the first power.

You could view this, and I like going with this definition as opposed to this one right over here.

If we were to make them consistent, if you were to make this definition consistent with this, you would say: hey let’s start with a 1 and then multiply it by 1 eight times, And you’re still going to get a 1 right over here.

But let’s do this with negative 1, So we’re going to start with a 1 and then we’re going to multiply it by negative 1 one time times negative 1, And this is of course going to be equal to negative 1. Now let’s take negative 1 and let’s take it to the second power.

We often say that we are squaring it when we take something to the second power, So negative, 1 to the second power.

Well, we could start with a 1.

We could start with a 1 and then multiply it by negative 1.

Two times multiply it by negative 1 twice And what’s this going to be equal to And once again by our old definition.

You could also just say: hey ignoring this one, because that’s not going to change the value.

We took two negative 1’s and we’re multiplying them Well negative 1 times negative 1 is 1 And I think you see a pattern forming.

Let’s take negative 1 to the third power.

What’s this going to be equal to? Well, by this definition, you start with a 1 and then you multiply it by negative 1, three times so negative 1 times negative 1 times negative 1 Or you could just think of it.

As you’re taking three negative 1’s and you’re multiplying it because this 1 doesn’t change the value, And this is going to be equal to negative 1 times. Negative 1 is positive 1 times negative 1 is negative.

1.

So you see the pattern Negative 1 to the 0 power is 1 Negative.

1 to the first power is negative 1.

Then you multiply it by negative 1.

You’re going to get positive 1.

Then you multiply it by negative 1 again to get negative 1 And the pattern you might be seeing is, if you take negative 1 to an odd power, you’re going to get negative 1 And if you take it to an even power, you’re Going to get 1 because a negative times a negative is going to be the positive And you’re going to have an even number of negatives so that you’re always going to have negative times negatives.

So this right over here.

This is even Even is going to be positive, 1, And then you could see that if you went to negative 1 to the fourth power Negative 1, the fourth power.

Well, you could start with a 1 and then multiply it by negative 1 four times. So a negative 1 times negative 1 times negative 1 times negative 1, which is just going to be equal to positive 1.

So if someone were to ask you, we already established that if someone were to take 1 to the I don’t know 1 millionth power.

This is just going to be equal to 1.

If someone told you let’s take negative 1 and raise it to the 1 millionth power.

Well, 1 million is an even number, so this is still going to be equal to positive 1.

But if you took negative 1 to the 999 999th power, this is an odd number.

So this is going to be equal to negative 1 .

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