Olympiad Mathematics | Indian | Can You Solve This?

Added: 2026-05-31

[00:00:01]Okay, if you're ready, let's solve this one quickly.

[00:00:07]We have the square root of 2 to the power of x minus 1 plus the square root of 2 to the power of x minus 3 and is equal to 12.

[00:00:23]So, how do we solve this problem here?

[00:00:27]It is simple.

[00:00:30]Because we know that if you have the square root of a, you know you can write it as um a to the power of 1 over 2.

[00:00:42]Okay? So, if this is true then look at what I would do to this equation here.

[00:00:49]We can write this as 2 to the power of x minus 1 and everything will be raised to the power of 1 over 2.

[00:00:58]Then, the same thing happens there. We have 2 to the power of x minus 3 and everything will be raised to the power of 1 over 2 as we have 12 on the other side.

[00:01:14]Now, we cannot add Okay? You know we cannot add um how do I say it? We cannot add this and this because they are not the same thing.

[00:01:26]But, if you open the bracket now, trust me, you're going to have um you're going to multiply both of them cuz that's the relationship between both of them.

[00:01:38]But, the power here is x minus 1, the power here is x minus 3.

[00:01:44]Let's say x minus 1 is the same thing as x minus 3 plus 2.

[00:01:54]Okay? Do you agree with that?

[00:01:57]x minus one is equal to X minus three plus two.

[00:02:01]I'm doing this so that we will have a common factor.

[00:02:05]So in place of this, I'm going to write this.

[00:02:08]So let's do that.

[00:02:10]And we will have two.

[00:02:14]Okay, this is two to the power of X minus three plus two.

[00:02:21]And the whole of this is raised to the power of one over two. Then we have our plus two to the power of X minus three.

[00:02:31]So to the power of one over two and it's still equal to 12.

[00:02:38]Okay, so guess what I will do from here.

[00:02:42]I'm going to open the bracket. So we have two to the power of X minus three plus two and we divide the whole of this by two. Then plus the same thing two to the power of X minus three and we divide the whole of this by two.

[00:03:05]And we have 12 on the right.

[00:03:09]Interesting, right?

[00:03:11]Now, how do we get this out of this?

[00:03:14]We're going to work on this again to get two to the power of X minus three.

[00:03:21]Close this and then we add two.

[00:03:24]Remember this is still over two. So we have not changed anything there. To go on, we have two to the power of X minus three over two.

[00:03:37]This is equal to 12. We're not changing anything here. We're not, you know, rearranging what we have here. We're only working on this.

[00:03:46]So you can see the idea of what I'm talking about. X minus three is here and it's going to be there.

[00:03:52]But there's something else you're going to do.

[00:03:55]Just write two to the power of x minus three.

[00:04:00]Let's have this over two.

[00:04:02]Then plus this is two over two as well.

[00:04:07]Do you think we have changed the power here? No.

[00:04:12]If you you know, the LCM of two and two is two. Then we're going to add the numerators. That will give you this.

[00:04:19]So we now have our plus two to the power of x minus three. This is still over two.

[00:04:26]And we have equals 12.

[00:04:30]So if we go on, we'll have two to the power of x minus three.

[00:04:38]This is over two. Then we have plus one.

[00:04:41]Then plus two to the power of x minus three over two.

[00:04:47]And this is equal to 12. Remember this two is the base.

[00:04:53]Right? And the other is um the exponent, right?

[00:04:58]Now, we can do something here using one of the laws of indices.

[00:05:02]The law that says a to the power of b plus c is a to the power b times a to the power c.

[00:05:12]You can remember that, right?

[00:05:14]So let me remove this.

[00:05:18]So I will apply the same thing to this.

[00:05:20]And we shall have two to the power of x minus three over two multiplied by the same two to the power of one from here.

[00:05:32]Then we have our plus two.

[00:05:35]Let me bring it down. We have our two to the power of x minus three over two.

[00:05:42]And this is equal to 12.

[00:05:45]So let's go on. Remember two to the power of 1 is the same as 2.

[00:05:52]Okay, so this is where we are.

[00:05:54]And um from here now, there's a common factor already, which is 2 to the power of x - 3 divided by 2. So, we bring out the common factor.

[00:06:07]So, we open bracket. If you divide the whole of this by this, you're going to have 2 to power 1 left. And 2 to power 1 is the same as 2.

[00:06:17]Plus, if you divide this by this, you're going to have 1.

[00:06:21]And this is equal to 12.

[00:06:24]So, what do we do?

[00:06:26]We have 2 to the power of x minus 3 over 2 multiplied by 3 to be equal to 12.

[00:06:35]Since we're trying to get the value of x, we're going to remove this 3 first as we divide both sides by 3.

[00:06:45]So, on the left-hand side, we will have 2 to the power of x minus 3 divided by 2 to be equal to 12 divided by 3, which is 4.

[00:06:57]And what is 4?

[00:07:00]4 is 2 to the power of 2. So, we're getting 2 to the power of x minus 3 over 2 to be equal to 2 to the power of 2.

[00:07:13]Now, the bases are equal. We're going to equate the powers.

[00:07:17]From one of the laws of um indices, right?

[00:07:21]Or one of the laws of exponents.

[00:07:24]Exponential equation, by the way. So, here we're going to have 2.

[00:07:29]Okay, we're letting go of the 2, so we equate the powers. x minus 3 over 2 is equal to 2.

[00:07:39]And this 2 can be over 1, so we cross multiply.

[00:07:42]And if we do that, we're going to have x minus three to be equal to two times two, which is four.

[00:07:51]Now, x is equal to four plus three.

[00:07:56]So, the value of x now is equal to seven.

[00:07:59]Now, let's put this value into the original equation very quickly.

[00:08:07]This is the equation, and our value of x is seven. So, if you put it here, you have the square root of two to power seven minus one plus the square root of two to power seven minus three. So, let's see if this will give us 12.

[00:08:27]So, this is the square root of two to power six plus the square root of two to power four.

[00:08:36]Now, what do we do?

[00:08:38]We remove the um the square root, so we have two to power six.

[00:08:43]Now, raised to power one over two from what I explained before.

[00:08:48]Then, we have two to power four then raised to the power one over two.

[00:08:55]Remember, I explained this before.

[00:08:59]And um two can go into six. We have two to power three.

[00:09:04]Plus um this two can go into this, and we have two to power two.

[00:09:10]And what does this imply?

[00:09:12]It means that we're having two to power three is eight.

[00:09:16]Two to power two is four.

[00:09:18]And the whole of this is equal to 12.

[00:09:21]Remember, we had um 12 on the other side of the equation.

[00:09:27]So, this means that our value of x which is equal to seven, satisfies the equation.

[00:09:36]Thank you for watching.