A nice math Olympiad algebra question.

a, b, and c in this particular equation—this is how we do it, guys. This is how we are going. to do it, so we are told that two to the power of a plus two to the power of b plus two to the power of The power of c is equal to 148. First of all, let's multiply and divide the left-hand side of this. Divide the equation by 2 to the power of a, which means we have 2 to the power of a in parentheses; let's use it. to multiply this two to the power of a plus two to the power of b plus two to the power of what C, yes, divided by what two to the power of A, all right, is equal to 148, all right. Now what do we do? Next, looking at this, we can now write this as two to the power of a bracket. what is two to the power of a over two to the power of a? Yes, right, since this two a two to the power Since a is more, we are dividing everything here, so we can write this as two to the power of a over two. two to the power of a plus two to the power of b over two to the power of a plus two to the power of c over two to the power of a, all right, now everything is equal to 148, what do we do next? Looking at this now, we can see that we have one here. so we can now rewrite this as two to the power of a bracket one plus two to the power of b over two to the power of (a plus two to the power of c divided by two to the power of a), everything is Equal to 148. All right, interesting. Now I want us to know something—we can rewrite this one for it. as let's just find factors of it, where one has to be or the other has to be, even so if we Divide this by two, we'll have 74. If we divide this by two, we have 37. All right, now we have an odd number. factor, and we have an even factor, so that means we can write one for it as two times two, which is four right four times 37, so let's go ahead and do that, but if this is the first time you're coming Once you've enjoyed this lovely and wonderful channel, it's time for you to click on that subscription. Click the Yes button on that subscription button and don't forget to turn on your notification bell Do you know why? Next time I'm going to upload a lovely video, you'll be able to watch it. All right, so let's solve this thing. Now we have two to the power of a bracket one plus 2^b / 2^a + 2^c / 2^ All right, is equal to. Now we are going to write this as what? Four times 37. Remember. Yeah, four dot words 37, please. This dot means multiplication. All right, notice that this side In the equation, 2a is an even number, and this side is odd. How do I know that if you test? They notice that if you call it, if you put it, even if you put odd here, you're still going to get even. Even if you put an even number here, you're still going to get an even result, so this is even if you also Test around this side; you're going to get odd. So now let's see, we are now going to have that 2a. is even, so see, this side is even, this side is odd, now this one is what, even, and this one is What odd? Okay, so let's equate even to even; it means that even is equal to the even side. We now have that two to the power of a is equal to the even side, which is what four, all right now. remember the way we solve this kind of thing using our law of indices, we can say that two to The power of a is equal to two to the power of two right now since they have the same base. Since they have the same base, we can simply equate the exponents, which means that a is equal to two. Yeah, we've found the value of the remaining a, b, and c, so stick with me, grab your popcorn, and don't move, and all right, so now let's solve the odd side of this equation when you look at this You know that one plus two to the power of b over two to the power of a plus two to the power of c 2 to the power of a is equal to 37. Okay, now let's subtract one from both sides of the equation. equation: if we subtract one from both sides of the equation, then we are going to have one minus one is gone, this one will be zero, right? So we'll be left with two to the b over 2 to the power of a plus 2 to the power of c over 2 to the power of a is equal to 37 minus one, right? Remember, we are subtracting one from both sides. so that seven minus one will give us 36 is equal to 36, all right? Permit me to Clean this side of the board. All right, now this is what we have now since both of them since they share the same denominator, we can actually rewrite this as what two to the power of b plus 2^c divided by 2^a is equal to what? 36. Yes, we can write it like this: This is okay; we can write it like this. Then let's go. Remember that we found the value of a to the All right, let me write it somewhere: a is equal to two, so we can now replace a with two here. so we now have two to the power of b plus two to the power of c divided by two to the power of two is equal to 36, so we can now rewrite this as 2 to the power of b plus 2 to the power of c. divided by four equals 36; now, when we cross-multiply, if we cross-multiply this particular the equation we will have is that two times b plus two times c, that is, two to the power of b plus two to the power of c is going to be equal to four dot what 36 now we have even we have even remember our trick I want to extract the odd number from these 36, so let's try that now. Two we divide 36 to give us 18. two we divide 18 to give us nine, so that means we can express that six as four times nine, so we we can now write this as 2 to the power of b plus 2 to the power of c nine okay so we can write this as what we can now write this as two to the power of b plus two to the power of c equals four times four, which is 16, so now we have one even and one odd All right, remember, stay tuned. Trust me, by the time you get the solution to this, you're going To enjoy it, you're going to be happy you did—you actually watched to the end. All right, now we are going to apply the same trick we applied before, which is what we are going to divide the left on the left side of the equation, we're going to multiply and divide this by two to the power of b, so we now have that two to the power of b times two to the power of b plus two to the power of c Everything divided by two to the power of what b is equal to 16.9. Okay, so this is what we have. Here's what we have here, so that means we can write this particular side as what? two to the power of b—now this is two to the power of b over two to the power of b, right? so it's going to be two to the power of b over two to the power of b plus this will be two to The power of c over two to the power of b, two to the power of c over two to the power of b, is equal to 16 times nine, all right. Now, we divide this to give us one, so we have two to the power The expression "b brackets one plus two to the power of c over two to the power of b is equal to 16.9 now" is ambiguous and appears to be poorly structured or incorrectly phrased. However, based on the likely intended meaning and standard mathematical notation, the most plausible interpretation is: Remember our trick. Note that this is even, so let's equate it to even, and this is odd, right? Let's set it equal to odd so we can say that 2b is equal to what 16 now this implies that two to the power of b is equal to now when you have this kind of thing, what do you do? All you need to do is ensure that you put them in the same base is as easy as that; now 16 can be written as two to the power of four two to the power of four, so when you look at this, once the bases are equal, you can actually equate their words, the powers; therefore, we can say that b is equal to four. We have gotten the Value of b, all right. So at this point, what do we do? Um, let's equate. Well, let's solve the odd side. for the odd side we will have that one plus one plus two to the power of c over two to the power b is equal to nine; now we are going to subtract one from both sides of the equation. and when we do that, we are going to have two to the power of c over two to the power of b is equal to nine minus one, which is eight. Now note that we can rewrite this using the law of indices, we all know that a to the power of x divided by a to the power of y, all right, when written as a to the power of x minus y times x minus y, okay? So we are going to apply this here; we are We are going to apply this here so that we can now write this to be equal to 2 raised to the power of c minus b. is equal to eight, all right, so we now have that two to the power of c minus know that our b is Four is equal to eight now we all know that at this point we can actually put this eight in base two, so we have two to the power of c minus four is equal to eight, which is the same as two to the power of three, all right, so now we have that we can equate the powers since they are the same in the same base, so we can write that c minus four is equal to three; okay, that means our c is equal to this four, we cross over to become plus four; that means our c is equal to seven. Therefore, thank you so much for watching because our a is two, our b is four, and our c is seven.