 This video is part 3 of exponent properties. Alright, for these next two properties I want to look at a couple patterns. So 3 cubed is 27. And 3 squared is going to be 9. 3 to the first is going to be 3. And 3 to the 0. Now let's come back here and look and see if we can figure out what was happening. 27 to 9, what did we do? We divided by 3. 27 divided by 3 is 9. What did we do from 9 to 3? 9 divided by 3 is 3. So if I take my 3 and divide it by 3, I'm going to get what? 3 divided by 3 is 1. Alright, now if I divide by 3 again, I'm actually going to have 1 third. And if I divide by 3 again, I'm going to have 1 over 9. So when I look at this, I have 3 to the negative n. 3 to the negative 1 was 1 over 3 to the first. 3 to the negative 2, we could say that this is 1 over 3 squared. Again, this is 1 over 3 to the first. So if I have 3 to the negative n, it should be 1 over 3 to the n. So we had two properties going on here. The 0 property was in red here. 3 to the 0, anything to the 0 will be 1. No matter what the base, it's always equal to 1. I could have a big, great, big fraction with about 10 things in it, 0 is on the outside of the parentheses, it would be 1. Negative exponents, where there are purple problems here, and it says B, no matter what the base, to the negative n is going to be 1 over B to the positive n. So what did we really do? Well, my base was B and it became 1 over B. So let's take the reciprocal, and then once I took the reciprocal, I started out with a negative exponent, and now I have a positive exponent. So I have to change the sign on the exponent. And I can change it because I took that reciprocal. That took care of the negative for me. All right, let's do the negative exponents. What's the reciprocal of 7? 1 over 7. And change that from a negative 3 to a positive 3, and 7 to the negative 3 is the same thing as 1 over 7 to the positive 3. Here I have a negative number, so I want you to be careful. This is 1 over negative 2, it's the reciprocal, and since it's in parentheses like that, I'm going to put it in parentheses here, and my negative exponent becomes a positive exponent. So the negative exponent did not change the sign on my number. That's a common mistake, but it doesn't have anything to do with the negative of my number. The only thing that negative does is flip my base. All right, what's the reciprocal of 1 fourth? 4. I've taken the reciprocal, so now I want to make this a positive exponent. So it's 4 to the fourth, and I do happen in that 64. All right, now what? If I do it like we've been doing it, this would be 1 over a to the fifth positive 5, and then on the bottom we'd have 1 over a, and it would be to the positive 2. And that's kind of ugly looking. It really means divide. So I could come back and say 1 over a to the fifth divided by 1 over a squared. And remember when you divide fractions, you flip the second one. This is to the fifth. Time, you multiply and then flip the second fraction, so now it's going to be a squared over 1, and I'm left with a squared over a to the fifth. It's the same base, so I can use my properties. But what I want you to notice right here, a to the negative 2 was on the bottom, and it ended up on the top. And my a to the negative 5 is on the top, and it ended up in the bottom with the positive exponent. So there's a shortcut that says where if I have a fraction and I have negative exponents in it, I just have to move that factor to the other part of the fraction. It was in the bottom, so I moved it to the top. It was in the top, so I moved it to the bottom, and then give it a positive exponent. But let's finish this problem out to get the real final answer. Same base dividing, so we would have a to the 2 minus 5, which is a to the negative 3, and we never answer with negative exponents, so we're going to have to flip and take the reciprocal, so 1 over a to the positive 3 is my final answer. What happens here? The key thing to remember with these is order of operations, especially those first two. Those are the ones that really matter when we're talking about our exponent properties. So we have to do grouping symbols first, if we can simplify inside, we need to simplify inside, and then we have to take care of exponents on the outside. So I can't do anything in this parentheses, and I can't simplify in this parentheses, so I've got my g done, but I can now take my exponent and distribute it. So this particular parentheses, or factor here, becomes 3 squared, and h to the negative 2 being squared. So really, 3 squared is 9, and h to the negative 2 times 2 would be negative 4. That's this first parentheses simplified. I still have to multiply it by the 4h to the third. That didn't change. I just haven't done anything with it, and now remember you take your numbers times themselves, and you take your variables times themselves. So 9 times 4, and then I'm going to have h to the negative 4 and plus 3. Same base, we're going to add the exponents. So final answer. 9 times 4 is 36. Negative 4 plus 3 will be h to the negative 1, and we really don't have a final answer here because we have to answer in positive exponents. So 36 is really over 1. So 36 is in the numerator. This is over 1, so I take it to the denominator and give it a positive exponent. Let's do this next one. Inside the parentheses, grouping symbols. Can't simplify anything. I have 1 number, 1b, 1a, nothing to simplify. So I've got that one done. Now I'm ready to do the exponent. And remember everything gets done to that negative 2. And remember again that my negative number is not going to change sign just because I have a negative exponent. It may change sign because I have an even exponent, a to the 6 to the negative 2. Multiply our exponents here. Negative 8 to the negative 2. And also on the top we have b to the negative 2. And simplifying the bottom, a to the 6 raised to the negative 2 will be a to the negative 12. And this is where that rule about fractions and negative exponents really works. This a to the negative 12 to make it a positive 12 has to go to the top. And this negative 8 to the negative 2 can come down to the bottom to give it a positive 2 exponent. And the b to the negative 2 can come down to the bottom to give it a positive 2 exponent. So the a to the negative 12 now is on top. And it's a to the positive 12. And I have negative 8 but now the exponent is to the positive 2. And I have b and its exponent is positive 2. So we're almost done. We have a to the 12. Negative 8 squared would be positive 64. And b squared. And finally, look at that thing. Well remember grouping symbols come first. So I can work inside my grouping symbols because I've got numbers and I've got a's and I've got y's. So inside here, 18 divided by 6 is going to be 3. And then this is a to the negative 4 minus 2. Remember you subtract your exponents when you're dividing. And then y to the 3 minus a negative 1. Again subtracting my exponents. And eventually that's all going to be taken to the negative 4. But right now I want to simplify my subtractions. So 3 negative 4 minus 2 will be a to the negative 6. And y to the positive 3 minus a negative 1. Remember you can add the opposite. So it's going to be y to the 4. And then that's going to be raised to the negative 4 power. Now I've done everything I can do. I've got a number 1a factor with an exponent and 1y factor with an exponent. So I'm ready to do my e. I'm ready to do the outside exponent. So 3 to the negative 4 a to the negative 6 to the negative 4 and y to the positive 4 raised to the negative 4. I can multiply these two. So I'm going to carry along my 3 to the negative 4. I want to get down to simplify exponents before I start moving things. a to the negative 6 times negative 4 would be a positive 24 and y to the 4 times negative 4 would be to the negative 16. Now I'm ready to move. All of these are in a numerator. So if I think about it being a numerator then the 3 to the negative 4 has to go to the bottom and the y to the negative 16 has to go to the bottom. But I don't do anything with my a to the 24 because it's positive. So I keep it where it is. So a to the 24 is on top. I have 3 to the fourth and I have y to the positive 16. And I do know what 3 to the fourth is. So a to the 24 over 81 y to the 16.