How to solve equations involving indices.

hi everybody, good evening. okay, today we are asked what are the values of m and n if 3 raised 2^(m - n) = 1 and (16^m) / 4 = 8^m 3m minus n, okay. In this particular question, now we are supposed to solve this. This is indices. We are supposed to solve this using indices, and we are going to get some fractions. equations, and we're going to use those equations to get the values of our m and our n. Okay, let's Get to it. Solution. Solution. Okay, this is the solution. Now, first of all, the first equation. says that 3 raised to the power of 2m minus n is equal to 1, okay, this one is a very easy one because we Everyone knows that any number raised to the power of zero is one, except zero itself. Any number that is raised Now these are indices; I want to balance their base, which means I can write that 3 raised to the power of 2m minus n equals 3 raised to the power of 0, since 3 raised to the power of 0 is also equal to 1. Okay, now that they have the same base, I can equate their exponents, so I can now say that This becomes our Equation One. Okay, this becomes our Equation One. Now, the next equation says that 16 raised to the power of m, all over 4, is equal to 8 raised to the power of 3m minus n, now looking At this point, since these are indices, I need to balance their base. Okay, so how do I balance their base? Yes, if I look at this, I know that I can express this as 2 raised to some power. 2 raised to the power of something and 2 raised to the power of something, so let me put all of in the same base because one thing with indices is that if you are able to put everything In the same base, you are definitely going to solve that question and get the solution very Alright, so let me do this here: 16 is the same thing as 2 raised to the power of 4, so I'll replace it. It's okay, all right, all over this 4 is the same thing as 2 raised to the power of 2. 2 raised to the power of 2 is equal to 8, which is the same as 2 raised to the power of 3. I hope you understood what I mean by 2 raised to the power of 3 being equal to 2 times 2 times 2 2 times 2 is 4, 4 times 2 is 8, so it means I can replace this 8 with 2 raised to the power of 3. Alright, so I'll now write this as 2 raised to the power of (3m minus n). Alright, let's go ahead and solve this. Okay, so now that I have this, there is this law of indices. This law of indices states that if you have x raised to the power of m over x raised to the power of n, that is, they have the same base. That you can write this as x raised to the power of m minus n. That is, you choose one of the bases and You subtract their powers. Okay, so let's do that to this now. Now we have the same 2 raised to the power of 4m here and 2 raised to the power of 2 here. So we're going to pick one as well. Okay, so I can now say 2 raised to the power of 4m minus 2. Okay, it equals 2. Let me expand this bracket. 3 times 3m gives us 9m. Minus 3 times n gives us 3m. Now, we have succeeded in expressing two of them with the same base. Okay, so if they have the same base, that means we can set their exponents equal. This implies that 4m -2 is equal to 9m minus 3m. Can you see that? Okay, I haven't gotten this yet. I'll try to put it in this form. Okay, so we can solve this using simultaneous equations. Now, looking at this now, I notice that my 2m and my n are on one side, then my digit, my ordinary number is on the other side. So let me start with this 4m. So I will bring this 9m to come and join this 4m. So I have 4 million. This 9 million is plus 9 million here, right? You know it's plus. Once there is no negative sign here, that means it is positive. So if it crosses this equation to come and meet this 4m, it becomes Minus 9m. The sign will change. So we are going to have minus 9m. Okay, now this 3n, remember that we have c, we also have n on this side. So I want this 3n to also cross over and come to this side. Okay, so this 3n is negative. If it crosses this equation, it becomes positive. So we are going to have plus 3n. Okay, it's equal to. Now this minus 2 that is here, we cross over to this side and if it crosses over, it becomes 2. Hope you understood, because this minus sign will change once it crosses the equals sign to the other side. Okay, so what do we have now? We have collected 9 terms. 4m minus 9m is going to give us minus 5 m. Okay, so we have minus 5 m plus 3n equals 2. So it is now in this form. Let's call this equation 2. So notice that we have succeeded in obtaining equation 1 and equation 2. We are now going to solve them simultaneously. So the first equation we obtained was what? 2m minus n equals 0. Okay, this is our Equation 1. Now let's write this as Equation 2: -5m + 3n = 2. This becomes our word, our Equation 2. Okay, now that we have this, what do we do? I want to eliminate one of these; what is it called? One of these variables. You know that we are looking for m and we are looking for n. For us to obtain either one, two of them should not be available. One should be available. So I need to remove this m so that I will have only n to work with, so that I can get my n. Now the only way I can remove this m is if This place and this place are the same. As in, I have 10, 10 here now. I can say 10 minus 10, this m will go. It becomes zero. I will now have only this to work with. And how do I do that? The only way I can achieve that is by swapping these coefficients. What do I mean? Now, I will multiply equation 1 by 5, and then multiply equation 2 by this 5. The coefficients of m in this 5, ignore the negative sign. Okay, just ignore the negative sign. So I'm going to multiply equation 1 by 5 and I'll multiply equation 2 by 2. Alright, I'll swap their coefficients and multiply. Just watch how this turns out. Five times two million now will give me ten million. Okay, minus 5 times n will give me -5n, which is equal to 5 times 0, which will give me 0. Do you understand? So this becomes equation 3. Now let's solve this one. 2 times minus 5n will give us Minus 10 meters. Okay, you can see that I've succeeded in making this 2 the same thing. Now, 2 times plus 3n will give us plus 6n, right, is equal to 2 times 2, which will give us 4. Let's call this equation (4). Now, since we have different signs, this one is negative, that one is positive. For two of them to cancel each other, we are going to add. Add equation 1 and equation 2. But assuming they are the same sign, let's say 10, 10, you know, when you have the same sign now, 10 and 10, subtracting 10 minus 10 gives you 0. Okay, but since they have different signs, This is positive and this is negative; the only way for two of them to cancel out is if you add them. Now you say 10 plus minus 10 is the same thing as 10 minus 10. Okay, so it's going to be what? 0. So this one has canceled out. See what I have: 10 plus minus 10. Plus times minus is always minus. Okay, so minus will always cancel the plus, so it becomes 10 minus 10 and we get 0. Now, minus 5n. plus 6n, minus 5 plus 6 is the same as if you write your 6 first, 6 minus 5, so it's going to Here, n. Okay, only one remains. Now, n equals 0 plus 4, which is 4. So you have succeeded. in obtaining the value of n. If this is the first time you are encountering my post, Please subscribe to the channel, okay, because that is the only way you will be notified anytime. I've uploaded another video. Alright, okay, let's continue. We've determined that n equals 4. So how do we get our m? We can get our m by substituting n = 4 into any of these equations. Okay, so let's say we substitute n equals 4 into, let's say, equation 1, okay, into equation 1. Do you Understood? Now, what is our Equation 1? We have that 2m minus, instead of us writing n here, We are going to write 4 is equal to 0. Do you understand? Okay, so 2m now, 2m minus 4 is equal to 0. Now, we have that 2m equals, this minus 4 will cross over to become plus 4. Right? Therefore, how do we get our m? Our m equals 4 divided by 2. Guys, If we say this over 2, this over 2, this 2, we cancel this 2, you are left with only m. So your m equals 4 divided by 2, 2 divided by itself is 1, 1 divided by 4, you are going to get 2. That means our m is equal to 2. Alright, have a lovely day and bye.