Olympiad Mathematics | The three solutions | Indian

Añadido: 2026-05-12

[00:00:01]Okay, let's get the complete solution to this equation here.

[00:00:07]Solution, we have m to the^ 3 to be equal to 30 to the^ 3 + 24 to the^ 3 + 18 to the power 3.

[00:00:28]Okay. So how do we solve this problem?

[00:00:31]Here we have m ^ 3.

[00:00:36]Now looking at this it's okay if you want to use calculator to you know express each of this but that is going to give you a very large or very large numbers right but we can simplify it without calculator. And this is what we are going to do. 30 to the^ 3 can be 6 * 5 to the^ 3.

[00:01:01]Then 24 is 6 * 4 to the^ of 3. Now in your head you can be saying why are you multip why why is 30 not 10 * 3? Why is 24 not 8 * 3? I'm doing this because I want to have a common factor and that is going to be six. Six is here. Six is here and I'm going to bring it out from 18 as we have 6 * 3. This is to the same power of 3. I hope you understand the explanation there.

[00:01:40]Okay.

[00:01:42]Now from one of the laws of indices we know that a m Okay. Okay, let me use AB.

[00:01:50]A B to the power of C. This is the same thing as A to the power C * B to the power of C.

[00:02:00]A ^ B a ^ C * B to the power C. So I'm going to apply this same um law to what we have on the right hand side.

[00:02:12]Right? Let me remove this. So we are going to have m ^ 3 now on the left to be equal to here we have 6 ^ 3 * 5 to power 3 okay then + 6 ^ 3 * 4 ^ 3 + 6 ^ 3 * 3 ^ 3 this is what we have and at the end of the day we have m ^ 3 to be equal to 6 ^ 3 is a common factor. Here we have 5 ^ 3. We have 4 ^ 3 remaining there. And here we have 3 to the power of 3.

[00:03:01]Now what do we do?

[00:03:03]By the way, if you're getting what I'm doing, then you have to subscribe. Yes, you have to. So we are going to get um to work on what we have in this bracket.

[00:03:15]So m ^ 3 is equal to 6 ^ 3 into bracket 5 ^ 3 is not 15 right because it is 5 * 5 * 5 and that is 1 2 5 4 ^ 3 is not 12 but 64 that is 4 * 4 16 16 * 4 and that is 64 and here we have 27 not 9 okay So m ^ 3 is equal to 6 ^ 3 * by the sum of 1 2 5 6 4 and 27 is 216.

[00:04:00]And luckily for us 216 is a perfect cube.

[00:04:05]Perfect cube because we can just find it um cube root.

[00:04:10]And um that means that we can write this as m ^ 3 = 6 ^ 3 into 21 6 is the same as 6 ^ 3 as well.

[00:04:22]Can see that this is very interesting.

[00:04:26]Now we will apply one of the laws we've already talked about. So our m^ 3 is equal to 6 ^ 3 right? Then we multiply by okay I think I'm going to combine something here. Let me do combination here. Okay, we have the same powers, right? So, we can multiply the bases and we get 36. That is 6 * 6. And this will now have the same power of three. I hope you understand what I did. You can pick one power and then multiply the bases or you pick one base and add the powers.

[00:05:06]Right? So let's remove this bracket because it's not needed anymore. So the next target is to bring everything to the same side probably the left hand side. So our m to the power 3 - 36 to the power 3 is equal to zero. And this gives us a um difference of two cubes.

[00:05:33]And we all know that if you have a cube minus b cube that it is simply equal to a - b as you multiply a 2 by a b + b 2 our difference of two cubes.

[00:05:54]Now our a is going to be m our b is 36.

[00:06:01]Then a squ is 30.

[00:06:04]Okay. Um a² is m 2. My bad. Okay. A² is m 2 then + a b. That's going to be m * 36. And it is 36 m. Then + b² which is 36 squared.

[00:06:24]36 squared. So we equate all of this to zero, right? And if you look at this, the product of these two is giving giving us zero, right? So that means that either of the factors here is going to be equal to zero. So let's do that very quickly.

[00:06:50]Okay. So we will now apply the zero product rule to say that m - 36 is = 0 or m² + 36 m + 36 2 is = 0. Now I didn't want to you know square this yet so that we not be dealing with big numbers. From the left hand side, our m is just 0 + 36 and that is 36.

[00:07:25]We have a solution already, but we need two more solutions from the other side.

[00:07:31]So, what do we do?

[00:07:34]What do we do? Let me bring it down here. We have m 2 + 36 m + 36^ 2 to be equal to 0. This is the quadratic equation and we can solve it by using the formula method. If you're using the formula, you're going to get a which is the coefficient of m squared and it's 1.

[00:08:01]You're going to get b which is the coefficient of m and is 36. then you're um you're going to get C which is the constant which is 36 squared. Take note of the square on C. Now to continue we will now write the quadratic formula which is m = - b + or minus we have b² - 4 a c / 2 * a interesting right so what we will do is to just put in the values of a b c into the equation uh into the formula here. So our m will be - 36 plus or minus b ^ 2 is 36 2.

[00:08:56]Then we have -4 * 1 * 36 2 because c is still 36 squar. So it has to come down here.

[00:09:07]Okay, I wrote out of sight a little. So it has to come down there. And we divide all of that by 2 * 1 so that we will continue to get m to be equal to - 36 plus or minus we have the square root of here there's something common to this two and that is 36 squared. Write your 36 squar open bracket here have 1. Here we have four because 4 * 1 is 4. We have four and this is all over 2 * 1 which will still give us 2.

[00:09:48]That will still give us two. Um let's continue from here.

[00:09:54]Right? So we get our m now to be - 36 plus or minus we have um 36^ 2 * - 3 that is 1 - 4 and it's all over 2.

[00:10:12]Now we can split what we have there.

[00:10:15]Yes, we can split what we have very quickly to get m = - 36 plus or minus<unk> of 36 2 * the<unk> of -3.

[00:10:30]This is all over 2 all over 2.

[00:10:36]Okay. Now we can easily find the square root of 36 squ because we know that square root and square are going to cancel each other but then minus 3 will lead us to a complex solution. Right? So let's deal with that from here. We are going to get our m to be - 36 plus or minus the square root and the square will go. So we have 36.

[00:11:07]Then we have to multiply by the square root of 3. Now the square root of negative here -1. Right? The square root of that is I. So we now divide this by 2. To continue with this we can rearrange to get m = - 36 plus or minus.

[00:11:27]We have 36 I then <unk>3 over there.

[00:11:33]Okay? because 36 * I is 36 I and we divide this by 2 and from here 2 can go into 2 into the two um numerators so that m will be 2 into 36 is um - 18 there's negative so it's going to be -8 then plus or minus 2 into 36 I that will be 18 I then we have <unk>3 So we have two in one solution and we are going to split what we have. We have m okay before now we have our m to be 36.

[00:12:16]That is our first solution. Then we have m again to be equal to -8 plus or minus oh we pick only the positive now. So we pick um 18 - 18 + 18 I <unk>3.

[00:12:36]Then we have our M3 which is the third solution to be equal to -8 - 18 I and we have <unk>3.

[00:12:49]So these are the three solutions to the equation. Thank you for watching. Make sure you subscribe, like, comment, and share.