Olympiad Mathematics | Russian | Can You Solve This?

Added: 2026-05-06

[00:00:00]Okay, if you're ready, let's [snorts] solve this problem here.

[00:00:06]Solution.

[00:00:10]Okay, so we have 9 to the power of X 9 to the power of X minus 1 over 3 to the power of X plus 1 equals 26.

[00:00:26]So, what do we do first?

[00:00:28]Let's work on the left-hand side.

[00:00:31]Remember 9 here is the same thing as 3 3 to the power of 2, right?

[00:00:39]Then we have X outside.

[00:00:42]Then minus 1.

[00:00:45]Then the denominator remains the same.

[00:00:49]And that is 3 to the power of X plus 1.

[00:00:55]All of this is equal to 26.

[00:00:59]Now, what do you think I'm going to do?

[00:01:02]I can change the position of the power there.

[00:01:04]Because if we have M to the power of A and we have B outside. Now, we can write this as M to the power of B and then we'll take A outside. This is very possible, right?

[00:01:21]Okay, so if it is possible that means that I can write what we have there as 3 to the power of X.

[00:01:30]Then I'll take 2 outside. Then we have minus 1 over 3 to the power of X plus 1.

[00:01:40]All of this is equal to 26.

[00:01:46]Okay, so this is where we are. But there's something else you would I would like you to see.

[00:01:52]From the numerator we can apply difference of two squares to the numerator.

[00:01:56]So, that will be give us 3 X squared minus 1.

[00:02:03]But this 1 is the same as 1 squared, so we can put a square on it.

[00:02:07]And this is over 3 to the power of X then plus 1.

[00:02:13]All of this is equal to 26.

[00:02:18]Interesting, right?

[00:02:19]And I know you know about your difference of two squares.

[00:02:24]That if you have A squared A squared minus B squared this gives A plus B times A minus B.

[00:02:37]Okay, I believe you know about this, right?

[00:02:39]Okay, let me remove that.

[00:02:44]Okay, so the same thing will will happen at the numerator. I mean, happen to the numerator.

[00:02:50]So, we're going to have 3 to the power of X plus 1 in one bracket.

[00:02:56]Then in the second bracket we shall have 3 to the power of X minus 1.

[00:03:03]So, remember that this is still over 3 to the power of X plus 1.

[00:03:10]And all of this is equal to 26.

[00:03:15]So, by now you should know what to do, right?

[00:03:18]Because we have this as a term. So, it can cancel this from the top.

[00:03:26]So, what is remaining at the top is 3 to the power of X minus 1 being equal to 26.

[00:03:37]And what do we do to this?

[00:03:40]We can you know collect like terms and this will go to the other side.

[00:03:46]So, we are having 3 to the power of X minus 1. You know, we want to take the 1 away, right?

[00:03:53]And I've already written the minus 1.

[00:03:55]So, for me to remove it I'll just add 1.

[00:03:58]So, this minus 1 will leave the left-hand side.

[00:04:03]And on the right we have 26 plus 1 cuz I added plus 1, right? So, it has to reflect on the right-hand side.

[00:04:12]Now, 3 to the power of X on the left-hand side is equal to 27 on the right.

[00:04:22]And we can now equate the bases.

[00:04:25]Because 3 to the power of X on the left is equal to 3 to the power of 3.

[00:04:32]3 to the power of 3 is what?

[00:04:34]Is um 27. Now, the bases are the same, 3 and 3. So, the powers must also be the same.

[00:04:42]So, our X is equal to 3. So, this right now is the solution to the problem.

[00:04:51]But you know how we always do it. We try to verify our work to make sure that we are correct. Take note, we have our X to be 3. Let's put this value into the original equation.

[00:05:07]Okay, so this is the given equation.

[00:05:10]Now, let's carry out our verification.

[00:05:16]Okay, we want to be sure that the left-hand side and the right-hand side will satisfy the given equation.

[00:05:25]We have 9 to the power of X. That means we have 3 um sorry, 9 to the power of 3. Then we have minus 1.

[00:05:35]Okay.

[00:05:36]Then um the denominator we have the same um uh what do we have?

[00:05:45]We have the same um 3 to the power of 3 plus 1.

[00:05:52]So, we want to see how this will give us um We want to see how this is going to give us 27 I mean, 26 on the right.

[00:06:03]We can do this without using calculator, right? What do we do? We break the 9 like we did before. So, this will give us 3 to the power of 2 then to the power of 3.

[00:06:18]Right?

[00:06:19]Then we have minus 1. And this is over By the way, we know 3 to the power of um 3, right? 3 to the power of 3 is what?

[00:06:29]27.

[00:06:31]Okay. You know what? Let's leave this so that we'll cancel out once we go to the other side. So, we write 3 to the power of 3 plus 1.

[00:06:40]Now, let's come down here, change the position like we did while solving.

[00:06:46]So, we'll have 3 to the power of 3 then to the power of 2 minus 1. 1 is the same thing as 1 squared.

[00:06:56]Take note.

[00:06:58]Then we divide by 3 to the power of 3 plus 1.

[00:07:05]Now, this will give us difference of two squares from the numerator.

[00:07:11]So, this means that we are going to have 3 to the power of 3 plus 1 in the first bracket.

[00:07:19]And in the second bracket we have 3 to the power of 3. Then we have minus 1.

[00:07:26]This is all over 3 to the power of 3 plus 1. Now, this verification is very helpful.

[00:07:34]If you were not able to understand the the methods I used the method I used while solving, you will get it from here.

[00:07:42]Because it's almost the same thing we are doing.

[00:07:45]But this time around we are dealing with all figures.

[00:07:49]Not variable anymore. Not variable.

[00:07:52]So, from here this canceled this.

[00:07:56]Like I explained earlier and we'll just have 3 to the power of 3 minus 1. 3 to the power of 3 is 27. So, we have 27 minus 1. And this is giving us 26.

[00:08:13]Right? And if you check very well, it is 26 we had on the other side of the equation. Let's look at that.

[00:08:21]Okay, see your 26 on the other side. So, this is to confirm that X equals 3 truly satisfies the given equation.

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