A Nice math OLYMPIAD exponential question, how to solve for 'm'

Welcome back to my channel. Let's solve this together. Today we are asked to solve for m. if three to the power of m minus two to the power of m equals 65, all right, um, this is the first The first thing we need to do is let's raise each of these to the Power of two over two. Yeah, I know it will sound a bit strange, but you're going to see why we're doing it. that 3m raised to the power of 2 over 2 minus 2m raised to the power of 2 over 2 is equal to 65 Alright, um, we all know that a to the power of m raised to the power of n using the law of indices It's the same thing as a to the power of m times n, okay? Yes, the same thing as a to the power of m. times n means that the positions of m and n can be swapped. Right now, you can write a to the power of n raised to the power of m. All right, so let's go to this; that will show. you that we can actually manipulate this side any way we want as long as we are going to get the same thing, so we can now write this as three to the power of m over two. Let me bring this two Then let me square this, because if you still use this to multiply it, you're going to get The sentence I have no relation to anything minus two to the power of we are going to do The same thing here squared is equal to 65. All right, if you look at this, notice that we have difference of two squares yes difference of two squares and this is what difference of two Squares says that a squared minus b squared is equal to a minus b in parentheses times a plus b. Okay, now in this case, this is our A and this is our B, right? So let's do that. Let's A. equals 3 to the power of m over 2, that is the first number here, 3 to the power of m over then b equals two to the power of m over two, all right? Now we can now substitute a and b. into this equation, all right, so we can now say that a squared minus b squared is equal to 65, okay, so what do we do next? That means that we can confidently say that a minus b times a plus b equals 65. Okay, we have two factors here; let's split this into two. Also, all right, now when we divide this 65 by 5, if we should divide 65 by 5, we will have, um, one more remainder one, we'll have 13, so that means that 5 times 13 is equal to 65, so let's represent 65, because we have (a minus b) multiplied by (a plus b) equals 5 times 13, okay. Let's continue now. When you look at this, when you look at this, you know that a plus b must be greater than a minus b, which means that a minus b should be equal to the smaller number a plus b should be equal to 13, okay? If you've watched this far, that means that you are actually enjoying what you're seeing, and I think it's time for you to click on that subscribe button and don't forget to turn on your notification bell, then most importantly Please share your thoughts on what I'm doing, and let me know if you know a more concise approach. Please don't hesitate to leave it in the comment section; I would appreciate that. Thank you so much. So let's continue. That means that a minus b is equal to what? Five. Now let's call this equation one. Remember that we are not looking for a, nor are we looking For b, rather we are looking for m, but this is a method that we will follow to obtain our m, okay. So, excuse me, we can now also say that a plus b is equal to 13, a plus b is equal to 13. And this is what Equation 2. All right, now let's continue. This is what a simultaneous equation is. Now we have a minus b equals 5 and a plus b equals 13; we can actually subtract the equations. 2 from equation 1, and we are going to get something, okay? All right, let's do that, so if We say 1 minus 2, that is equation 1 minus equation 2, then we are going to have a minus a, we give all. Zero, right? So we are going to ignore that. Minus b minus plus b is the same thing as minus b minus b. which is what minus 2b, right? So remember, we are subtracting, so minus b minus plus b is minus b minus b, which is minus 2b, so we have minus 2b. Okay, is equal to 5 minus 13. We give By minus 2 minus 2, we cancel minus 2, and we will have that b is equal to negative; we cancel negative. Alright, then 2 we divide 8 to get 4, so that means our b is equal to 4. Okay, let's Let's set that aside for now; our b is equal to 4. We are going to return to that later. So let's call this equation. 3b equals 4, okay. Now you know how we solve simultaneous equations when you've gotten a solution for one of the variables; you now substitute what you got into any of these equation to find the other variable, okay, we have obtained the value of our b, so let's substitute b Substituting 4 into equation 1, you can still substitute into equation 2, and it will give you the same result. So if we do that, we will have that a minus 4, that is 4, b is 4, is equal to 5. now if we add 4 to both sides of the equation because we want a to be by itself on the right side, so we have a minus 4 plus 4 is equal to 5 plus 4, so minus 4 plus 4 is 0, so we have that a is equal to 9, so we have found the value of b and the value of a. But recall, recall, look at this. recall that our a is equal to 3 raised to the power of m over 2, I would like you to stick to the end of this video because we are going to check, and that is we are going to put the values that We got for m to see if we are actually correct, we are going to prove the solution at the end. All right, so permit me to clean this side of the board so that we can continue. So let me also clean. Here is my reasoning; let's just leave the most important equation here, which is this, okay all. Right now we have that a equals 3 to the power of m over 2. Now we have our a equal to 9. soością now say that 9 is equal to 3 to the power of m over 2. kuruluş of indices, you know that the best way you can solve any exponential is if you are able to put the Both numbers in the same base, so let's put this in base 3 as well, so we have that 3 2 to the power of 2 is equal to 3 to the power of m over 2, all right, since the bases are equal then the powers are equal as well, so we can now say that 2 is equal to m over 2. Okay, so if we cross multiply m times when we are given that m is equal to 2 times 2, which gives us 4, so we have found the value of m using what a; now let's use b as well. Recall that b is equal to 2 to the power of m over Now, what did we get as our b? 4. All right, so we now have 4 equals 2 to the power of m over 2. Similarly, just as we did here, we're going to do what makes this 4; let's put it in base. So it's going to be 2 to the power of 2 equal to 2 to the power of m over 2, okay. The bases are equal, so we can now equate the exponents. We can now write that 2 is equal to m over 2. If we cross-multiply, we have that m times 1 gives us m, which is equal to 2 times 2, which gives us 4. So notice that we are still going to get the same 4. Now let's check the results to see if m is actually equal to 4 now the question says that um this is what the question says that 3 to the power of m -2 to the power of m equals 65, okay, so we said that m equals 4, right? So we have 3. 3 to the power of 4 minus 2 to the power of 4 equals 65. Now, 3 to the power of 4 means 3 times 3 times 3 times 3, and if you do that, you are going to get 81, so this is 81 minus the same for 2 times 2 to the power of 4 gives us 16, which is equal to 65. Now, is this actually equal to 65? Let's see. 11 minus 6 gives us 5, 7 minus 1 gives us 6, so 65 is equal to 65, which means that our solution That's what's right, thank you so much for watching. Do have a great day, bye.