How to solve fractional and negative indices.

Hello everybody, welcome to my channel. Let's get straight to it. Okay, all right. So you're giving. something like this—this is under indices. The first thing you need to do is to look out for the bases and try to put them in the same base, that is, if you can put all of them in the same base three, so looking at this, notice that everything here is in base three, three, three, three, except This one, so I need to convert this into base three now. How do I do that? How do I know if it's even? if I can even represent it in base three, you divide by three—okay, just pay attention and say I'm going to analyze this question so that you can understand it very well—just follow me to the end. Okay, so I want to know if there's anything normal. I know that this is three raised to the power of Of three will give me 27, but assuming you're guessing, this is how to be sure about it. Divide by three because everything is in base three; three divides 27. Nine divided by three is three. Three divided by three is one. We give you three, we divide by three again, three divided by three will give you what? one you notice that three raised to the power of three that is one two three three times three times three, we also give you 27, okay, so that means that three raised to the power of three will give you 3³ so I'm now going to rewrite this and replace this 27 with what three raised the power of three, so that everything here will be in base three. All right, let me do that. So I'm not going to write this is our solution, okay? This is our solution. So our question will look like this: three raised to the power of n minus three raised to the power of n minus one all over three raised to the power of three times three raised to the power of n minus three raised to the power of three, so I'll now replace this 27 with what three raised to the power of three x times 3 raised to the power of n minus 1. Do you understand? Okay, that is the first thing to do. Now, if you look at this, the next thing we should examine now is the laws of indices that we are going to apply here. So follow me, let me show you the laws of indices that we are going to apply here. Now the first law of indices that What we are going to apply here is the multiplication law. Now what does the The multiplication law states that x raised to the power of a times x raised to the power of b is the same. thing as x raised to the power of a plus b. Now what does this mean? Let me bring it home. For example, if you have a number, if you have something like 2, let's say 2 raised to the power of 3 and you are multiplying it by 2 raised to the power of anything else is equal to, you are going to choose one of these two, you choose one of them, you Add the powers, that is going to be 3 plus 4. Do you understand? So that is what this law is about. is about. Once you have two numbers and they are in the same base, all you All you need to do is pick one of them, whether they are multiplying or not. When multiplying and they are in the same base, you choose one of them and add their exponents. 3 raised to the power of 2, raised to the power of 4. That is the same as 2 raised to the power of 3 minus 4. Okay, so that's the two laws I know you're going to use here. Okay, so let's go ahead and do that. Alright, but if you've followed me this far, I know you understand or you love what I'm doing—please, it's time for you to click that subscription button because I'd like to Okay, and the only way to do that is to combine their powers, and combine their powers. That's the only way I'll be able to bring the two together so that, if I factor them out, I'll end up with something Okay, so let's continue. Okay Then, if we factor it out here, we are going to have 3 raised to the power of n. Okay, if 3 raised to the power of n is divided by 3 raised to the power of n You're going to have one, right? If 3 raised to the power of n times 3 raised to the power of minus 1 is divided by 3 raised to the power of n This will cancel this. Your left switch only works with 3 raised to the power of minus 1. All over For this one, I want to factor out 3 raised to the power of 3 plus n because that is what they have in common. Okay? 3 raised to the power of 3 plus n, 3 raised to the power of 3 plus n So I'm going to factor out 3 raised to the power of 3 plus n I will open my bracket if this 3 raised to the power of 3 plus n is divided by 3 raised to the power of 3 plus n you are going to have 1 right minus if this 3 raised to the the power of 3 plus n times 3 raised to the power of minus 1 is divided by this, notice that This we cancel, this your left is only what 3 raised to the power of what minus 1, okay Are you seeing what I'm seeing? Okay, just look at them very carefully. Do you notice that we have 1 minus 3 raised to the power of minus 1, 1 minus 3 raised to the power of minus 1 in the bracket, so what do we do? It means they can be canceled out because of what we have Here's what with multiplication, so this can cancel this, so we are now left with what 3 raised to the power of n over 3 raised to the power of 3 plus what n okay now this is where we apply our division law, our division law, because we have the same base and this raised to the power of what n and 3 plus n, what do we do? We subtract their powers; we pick one three. and we subtract their power, so we are going to write 3 raised to the power of n minus We have 3 plus n in the denominator, but I'm going to put this 3 plus n in parentheses. Because I have a minus sign here, this negative sign is going to affect everything here. That's why I had to put it in brackets; if this were plus, I wouldn't have changed it. wouldn't have put it in brackets at all, okay? Now see how it's going to affect it now, this is equal to 3 raised to the power of n minus minus times 3 gives us minus 3 minus times plus Can you see that? Just stick with me and see how we're going to get the final result. Now this is equal to n minus n, which gives us zero, right? So you notice that everything here is It's raining with 3 raised to the power of minus 3. Now, what is 3 raised to the power of minus 3? Is sentience 1 over 3 raised to the power of 3? That is, if I want to remove this negative sign It is going to be 1 over 3 raised to the power of 3. Now, what is 3 raised to the power of 3? This sentence has 1 over 3 times 3 times 3, which is sentence 1 divided by 3 times 3 equals 9. 9 times 3 is 27, and this becomes our answer. Thank you for watching, bye.