Olympiad Mathematics | Indian | Can You Solve This Nice One?

Añadido: 2026-05-23

[00:00:01]If you're ready, let's provide a solution to this problem here.

[00:00:08]This is x to the power 4x - 12 = 32 to the power of 12.

[00:00:20]Okay, so how do we solve this problem?

[00:00:23]Just look at it.

[00:00:24]What do you think you can do?

[00:00:27]4 is here. 4 is a factor of 12, so you factor out the the 4. So, we're going to get x to the power 4, then multiply by x - 12 / 4 is 3.

[00:00:42]And this is equal to 32 to the power of 12.

[00:00:46]Okay, so you look at it again, you discover that 4 here can easily divide 12, right?

[00:00:53]So, we have to remove 4 from here.

[00:00:56]And for us to do that, we're going to do this.

[00:01:01]x - 3 then the whole of this will be raised to the power of 1/4.

[00:01:10]You see that?

[00:01:11]And then here we have 32 to the power of 12.

[00:01:15]The whole of this will be raised to the same power of 1/4.

[00:01:19]So, that this can take this out.

[00:01:23]And on the left-hand side, we will just have x to the power of x - 3.

[00:01:32]And it's equal to on the other hand, remember the relationship between these two powers, 12 and 1/4, is multiplication.

[00:01:41]And because of that, 4 can go into 12 three times. So, we are having 32 to the power of 3.

[00:01:51]Okay, so we have 32 to the power of 3.

[00:01:55]And what again do you think we can do?

[00:02:00]Okay, let's do this.

[00:02:01]We have x to the power of x minus 3.

[00:02:07]And then here we have 32 to be the same as 2 to the power of 5.

[00:02:14]Then we have power of 3.

[00:02:17]And like I told you, the relationship between the two is multiplication. So, that means we can change the position.

[00:02:25]Right? Because multiplication you know, um does not have to do with position.

[00:02:32]Right? For example, 3 * 2 is the same thing as 2 * 3.

[00:02:39]Okay, I believe you agree with me, right? So, let me remove this.

[00:02:44]And I will do the same to 5 and 3.

[00:02:48]x to the power of x minus 3 will now be equal to 2 to the power of 3. Then I will take 5 out.

[00:02:57]See that?

[00:02:59]Take 5 out. So, that from here we have x to the power of 3.

[00:03:05]By the way, this is um x to the power of x minus 3 to be equal to 2 to the power of 3 is what?

[00:03:13]2 to the power of 3 is 8. So, we have 8 to the power of 5.

[00:03:18]This is what we we have, right?

[00:03:21]Then if you look at the left hand side, we can apply this law of indices that says um m to the power of a minus b is the same as m to the power of a times m to the power of minus b.

[00:03:38]This is one of the laws of indices.

[00:03:43]Okay, let me remove this law.

[00:03:47]Okay, so now what do we have? We're going to have x to the power of x multiply by x to the power of 3.

[00:03:55]Now, this is equal to This is -3, right?

[00:04:00]This is -3.

[00:04:04]Now, we look at the other side of the equation.

[00:04:08]8 to the power of 5. Can we express this in this form? The answer is yes.

[00:04:13]Imagine we have 8 to the power of 8 cuz we already have the base as the power.

[00:04:18]So, 8 to the power of 8 multiply by Here again, we have the same base, so we are going to have 8.

[00:04:26]Then, what do you take out of 8 to get 5?

[00:04:31]It is 3, right? So, we are going to take minus 3.

[00:04:37]Now, look at what we just did.

[00:04:39]If you pick one of the bases here, you're going to have this 8. Then, 8 + -3 will give you this 5.

[00:04:48]So, this is the same as 8 to the power of 5.

[00:04:53]Now, we can relate very easily.

[00:04:56]This one should be equal to this and this should be equal to this.

[00:05:02]From this part, 8 to the x to the power of x should be equal to 8 to the power of 8. From here, what do we do? We can conclude that x is equal to 8.

[00:05:17]As easy as that. Then, if you go to the second part, you're going to have x to the power of minus 3 to be equal to 8 to the power of minus 3.

[00:05:28]And from one of the laws of indices as well, if you have the same powers, you can equate the base. So, our x here again is also equal to 8.

[00:05:42]So, what are we going to do?

[00:05:44]We conclude that x is equal to 8.

[00:05:48]And you know how we do it. We will not stop here. Let's go and put it back into the equation so that we can be sure.

[00:05:56]Okay, so this is our um original equation and the value of X we got from calculation is 8. So, we're going to put in the value of X now.

[00:06:06]We have X to be 8 to the power of 4 multiplied by by X X is still 8. Then we have minus 12. Remember on the other hand, we're trying to look for 32 to the power of 12.

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

[00:06:27]This is 8 to the power of 4 * 8 is 32.

[00:06:32]And we have minus 12.

[00:06:35]Now, let's see if this is going to be equal to 32 to power of 12.

[00:06:42]What do we do? 32 minus 12 is 20. So, we have 8 to the power of 20. And this is equal to 32 to the power of 12.

[00:06:51]Now, you look at this.

[00:06:53]Do you think the left-hand side will be equal to the right-hand side?

[00:06:58]Yes. Do you think it will be equal?

[00:07:01]If you ask me, I will say yes without using calculator. Let's try to make them have the same base.

[00:07:08]8 is 2 * 2 * 2, so that is 2 to the power of 3.

[00:07:13]And then we have 20 there.

[00:07:16]Okay, we have 20 over there. Then here we have 32, which is 2 * 2 * 2 * 2 in five places. So, we will write that as 2 to power 5.

[00:07:27]And then we have 20 I mean 12 here.

[00:07:31]Okay.

[00:07:32]So, if we go on, we can, you know, open the bracket as we multiply the powers.

[00:07:38]We have 2 to the power of 3 * 20 is 60.

[00:07:43]Then on the other hand, we have 2 to the power of 5 * 12 is also 60.

[00:07:50]So, what are you seeing?

[00:07:52]Left-hand side equal to the right-hand side. So, the value of X is truly equal to 8.

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