Negative exponents

We already know that 2 to the fourth power can be viewed as starting with a 1 and then multiplying it by 2 four times. So let me do that. So times 2, times 2, times 2, times 2. And that will give us, let's see, 2 times 2 is 4, 8, 16. So that will give us 16. Now I will ask you a more interesting question. What do you think 2 to the negative 4 power is? And I encourage you to pause the video and think about that. Well, you might be tempted to say, oh maybe it's negative 16 or something like that, but remember what the exponent operation is trying to do. One way of viewing it is this is telling us how many times are we going to multiply 2 times negative 1? But here we're going to multiply negative 4 times. Well, what does negative traditionally mean? Negative traditionally means the opposite. So here this is how many times you're going to multiply. Maybe when we make it negative this says, how many times are we going to, starting with the 1, how many times are we going to divide by 2? So let's think about that a little that. So this could be viewed as 1 times, and we're going to divide by 2 four times. Well, dividing by 2 is the same thing as multiplying by 1/2. So we could say that this is 1 times 1/2, times-- let me just do it in one color. So 1 times 1/2, times 1/2, times 1/2, times 1/2. Notice multiplying by 1/2 four times is the exact same thing as dividing by 2 four times. And in this situation this would get you, well 1/2, well 1 times 1/2 half is just 1/2, times 1/2 is 1/4, times 1/2 is 1/8, times 1/2 is 1/16. And so you probably see the relationship here. If you're-- this is essentially you're starting with the 1 and you're dividing by 2 four times. You could also say that 2-- I'm going to do the same colors-- 2 to the negative 4 is the same thing as 1/2 to the fourth power. Let me color code it nicely so you realize what the negative is doing. So this negative right over here-- let me do that in a better color, I'll do it in magenta, something that jumps out. So this negative right over here, this is what's causing us to go one over. So 2 to the negative 4 is the same thing, based on the way we've defined it just up right here, as reciprocal of 2 to the fourth, or 1 over 2 to the fourth. And so you could view this as being 1/2 times 2 times 2 times 2, if you just view 2 to the fourth as taking four 2's and multiplying them. Or if you use this idea right over here, you could view it as starting with a 1 and multiplying it by 2 four times. Either way, you are going to get 1/16. So let's do a few more examples of this just so that we make sure things are clear to us. So let's try 3 to the negative third power. So remember, whenever you see that negative, what my brain always does is say I need to take the reciprocal here. So this is going to be equal to, I'm going to highlight the negative again, this is going to be 1 over 3 to the third power. Which would be equal to 1/3 times 3 times 3, or 1 times 3 times 3 times 3, is going to be 27. So this is going to be 1/27. Let's try another example, I'll do two or three more. So let's take a negative number to a negative exponent, just to see if we can confuse ourselves. So let's take the number negative 4, and let's take it-- I don't want my numbers to get too big too fast. So let's just take negative 2 and let's take it to the negative 3 power. I'll make my negatives in magenta, negative 3 power. So at first this might be daunting, do the negatives cancel? And that will just be the remnants in your brain that are trying to think of multiplying negatives. Do not apply that here. Remember, you see a negative exponent, that just means the reciprocal of the positive exponent. So 1 over negative 2 to the third power, to the positive third power. And this is equal to 1 over negative 2 times negative 2 times negative 2. Or you could view it as 1 times negative 2 times negative 2 times negative 2, which is going to give you 1 over negative 8 or negative 1/8. Let me scroll over a little bit, I don't want to have to start squishing things. So this is equal to negative 1/8. Let's do one more example, just in an attempt to confuse ourselves. Let's take 5/8 and raise this to the negative 2 power. So once again, this negative, oh I got at a fraction is a negative here. Remember this just means 1 over 5/8 to the second power. So this is just going to be the same thing as 1 over 5/8 squared, which is going to be the same thing-- so this is going to be equal to-- I'm trying to color code it, 1 over 5/8 times 5/8, which is 25/64. 1 over 25/64 is just going to be 64/25. So another way to think about it is, you're going to take the reciprocal of this and raise it to the positive exponent. So another way you could have thought about this is 5/8 to the negative 2 power. Let me just take the reciprocal of this, 8/5 and raise it to the positive 2 power. So all of these statements are equivalent. And that would have applied even when you're dealing with non-fractions as your base right over here. So 2, you could say well this is going to be the same thing. 2 to the negative 4 is going to be the same thing as taking my reciprocal. So this is going to be the same thing as taking the reciprocal of 2, which is 1/2 and raising it to the positive 4 power.