Olympiad Mathematics | Russian | Can You Solve This One?

追加: 2026-05-27

[00:00:00]Hi everyone.

[00:00:04]Let's solve this one very quickly. This is 16 to the^ of m + 4 to the^ m = 36 to the^ of m.

[00:00:17]So what you will do is to look at what you have. Look at it and see if there's something you can do.

[00:00:25]Mind you, you cannot add 16 m and 4 m.

[00:00:29]You can't add but we can simplify the base.

[00:00:34]The three of them are perfect squares.

[00:00:36]So we can write this as 4 squ right then to the power of m plus there we can write that as 2^ 2 remember to the^ m and 36 6 2 the power of m.

[00:00:55]And then when you discover that you are having two powers in this form, you can easily easily multiply the powers. Okay.

[00:01:05]So we have 4 to the^ 2 m + 2 ^ 2 m and is equal to 6 ^ 2 m.

[00:01:18]Now I'm looking at this now. What do I do? What if I divide all through by 6 to the^ 2 m?

[00:01:27]Yes. So I'll divide this by 6 to the^ 2 m. Divide it by 6 to the^ 2 m and this by 6 to the^ of 2 m. So that on the right hand side we'll have one. So we are having our 4 to^ 2 m / 6 to the^ 2 m + 2 ^ 2 m / 6 to the^ 2 m and this is all equal to 1.

[00:02:01]This is all equal to one.

[00:02:05]Right? So from here now what do we do?

[00:02:10]We can do something. You can see that we're having the same power numerator and the denominator. And there's a law again that says that a to the power b over um c to the power b this is the same as a / c both to the power b. This is one of the laws of indices.

[00:02:34]So we shall apply this to what we have on the left hand side. So that we can have um 4 / 6 to the^ of 2 m plus there we have the same thing 2 / 6 but then to the power of 2 m and this is equal to 1. Okay. To solve this problem, we'll have to think at some point.

[00:03:03]We'll have to to think very well at some point. Now, let's reduce 4 / 6 to it lowest term. 2 into 4 is 2. 2 into 6 is 3. This is to the power of 2 m.

[00:03:19]2 into itself is 1. 2 into 6 is 3.

[00:03:23]Okay. And this is to the power of what?

[00:03:26]2 m.

[00:03:28]and all is equal to one. And this is the point that we will have to think.

[00:03:35]We have the same powers. But having the same power does not mean you can add the base, right? And it equally does not mean you should multiply the base. So what should we do? Here we have one and here we have 3 / 2 to this power. 1 / 3 to this power.

[00:03:56]Let's pick out the base and look at it.

[00:03:58]The base on the left is 2 over 3 and this base is 1 / 3. Imagine I want to add the two of them. What will I have?

[00:04:06]The LCM is three. So, I'll pick three here.

[00:04:12]Okay. Then 3 / 3 is 1. 1 * 2 is 2.

[00:04:19]This addition is here. 3 / 3 is 1. 1 * 1 is 1. And at the end of the day we have um 3 over 3 which is equal to 1. So if we add the bases here they are going to be equal to one. Now let me remove this again.

[00:04:44]So what we'll do now in place of this one we're going to add these two base and these two base numbers and leave them there. So let's write this 2 over 3 to the^ 2 m + 1 / 3 to the^ of 2 m. This is equal to 2 / 3 + 1 / 3.

[00:05:11]Okay. Since this will still give us one.

[00:05:14]So that if we have done this, the next thing we're going to do is to compare, right? Is to compare. If you like, you can write it in this form. This is 2 over 3 to the power of 2 m + 1 / 3 to the power of 2 m and is equal to 2 / 3 to the power of 1 + 1 / 3 to the same power of 1. So this way it will be very easy. This is addition. it will be very easy for us to compare.

[00:05:56]Okay. So from here we are going to compare but ordinarily we're not supposed to write this power of one but we need the one that's why I wrote the ones. So we compare these two.

[00:06:08]So 2 / 3 to the^ 2 m should be equal to 2 / 3 to the power of 1. So if we compare this since the bases are the same 2 m will be equal to 1 and we will divide both sides by 2.

[00:06:32]So that m will just be equal to 1 / 2.

[00:06:36]From the first case we got m to be 1 / 2. Let's go back to the other case. I think we should do it here. The other case is to compare the second um to compare this right here which is 1 / 3 to the^ 2 m to be = 1 / 3 to the power of 1. So if we do this the bases are the same. So 2 m should be equal to 1 as well. So our m will be 1 / 2 if we divide both sides by two. So definitely we can conclude that the value of m is 1 /2. But we still have to put this into the real the original equation. The equation is um what is the equation again? 16 to the^ m + 4 ^ m being = to 36 to the^ of m. So we have 16 ^ 1 / 2 + 4 ^ 1 / 2 = 36 or let's okay let me just write will it be equal to 36 to the^ 1 / 2 16 ^ 2 is the same as 16 to the^ 1 /2 is the same as the roo<unk> of 16 plus the roo<unk> of 4 will it be equal to the square root of 36? The answer is yes because square roo<unk> of 16 is 4 plus the square roo<unk> of 4 which is 2 and the square roo<unk> of 36 is 6. 4 + 2 is 6. So this means that our value of m is definitely 1 / 2. Thank you for watching.