Olympiad Mathematics | Japanese | Can You Solve This One?

Added: 2026-05-22

[00:00:00]Okay, if you're ready, let's provide a solution to this one right here.

[00:00:07]Solution.

[00:00:09]We have m to the power of 3 minus 18 to the power of 3 minus 24 to the power of 3 minus 30 to the power of 3 equals 0.

[00:00:27]Okay, and we want to solve this completely. What do we do?

[00:00:32]Um if you're going to press 18 to the power of 3, 24 to the power of 3, they're going to give you very large numbers, right?

[00:00:42]So, what do we do? Let's break them.

[00:00:45]We have m to the power of 3 minus 18 here is um 6 multiplied by 3.

[00:00:52]This is to the power of 3.

[00:00:54]Minus 24, which will be 6 multiplied by 4 to the power of 3.

[00:01:02]Then minus 30, which is going to be 6 multiplied by 5 to the power of 3. This is all equal to 0.

[00:01:12]The question that should be on your mind right now is why is 18 not 9 * 2?

[00:01:20]Why is 24 not 8 multiplied by 3?

[00:01:24]Okay, I'm using 6 because I want to have a common factor.

[00:01:29]Okay. Then from this particular law, a b to the power of c is the same as a to power c times b to same power of c.

[00:01:43]So, from this law, I'm going to express these three in this form.

[00:01:51]Okay, so that we can have m to power 3 minus 6 to power 3 times 3 to the same power minus 6 to power 3 times 4 to the same power minus 6 to power 3 times 5 to the same power and everything is equal to 0.

[00:02:14]Now, if I bring this m to the power of 3 you know, I can factor out 6 to power 3 since it's here, it's here, and it is here.

[00:02:24]So, 6 to power 3 is out as a common factor.

[00:02:28]What will be left over here is 3 to the power of 3.

[00:02:32]And again, you have to take note of this negative.

[00:02:35]The negative here, this this this is, you know, out here in the form of this.

[00:02:42]So, every other thing you're going to have will be positive.

[00:02:45]plus Um here we'll have 4 to power 3 plus there we have 5 to the power of 3.

[00:02:55]And we equate all of these to 0.

[00:02:59]The next step is that we work on these um terms.

[00:03:04]m to power 3 minus 6 to power 3 into 3 cubed is 9 is 27.

[00:03:12]4 cubed is going to be 64. That is 4 * 4 * 4.

[00:03:17]And the 5 cubed is um 125.

[00:03:23]This is all equal to 0.

[00:03:27]So, that from here we have our m to the power of 3 minus 6 to power 3 The addition of this is going to give us 216.

[00:03:40]And it's all equal to 0.

[00:03:44]But when you look at 216 216 is a perfect cube.

[00:03:50]Okay?

[00:03:52]It is a perfect cube.

[00:03:54]And it's um 6 * 6 * 6. So, that means we're having 6 ^ 3 again.

[00:04:01]Which is equal to 0.

[00:04:04]From one of the laws of indices, we can multiply the bases here and let them have the same power. So, we have our m ^ 3 from the left-hand side minus 6 * 6 to the power of 3. This is equal to 0.

[00:04:25]So, that our m ^ 3 minus 36 Okay, there may be no need for the bracket, so we put 3.

[00:04:36]This is equal to 0.

[00:04:39]And at this point, somehow, we have difference of two cubes.

[00:04:45]And if you have a³ - b³, this is the same as a - b into a² plus ab plus b².

[00:05:02]Okay, so this is what we call the difference of two cubes.

[00:05:09]Now, our a is m and our b is 36. So, we're going to put them into this expression now.

[00:05:16]a - b becomes m Right? Becomes m. Okay, let me write this one.

[00:05:25]Becomes m minus 36 Right?

[00:05:33]Then, into bracket, a² is going to be m² plus ab. That will be m * 36.

[00:05:43]And it's 36m.

[00:05:45]Then, we have plus b², which is going to be 36 squared.

[00:05:53]We close this. Remember that everything here will be equal to zero because of that particular zero.

[00:06:01]Uh from here we are going to apply our zero um zero product rule because we are multiplying both of them to get zero. So, either of them will be equal to what? Zero.

[00:06:14]Okay, so let's apply this.

[00:06:17]Okay, so applying the zero product rule, our m minus 36 is equal to zero or m squared plus 36 m plus 36 squared is equal to zero. This is a linear equation and this is a quadratic equation.

[00:06:39]From the linear part, our m will be zero plus 36 and the value of m is equal to 36.

[00:06:49]By the way, we have one of the solutions already.

[00:06:54]We have this as a solution. And I'm going to bring down this quadratic equation and then we will solve it.

[00:07:02]The equation again is m squared plus 36 m plus 36 squared equals zero.

[00:07:12]Do not bother expanding this 36 squared.

[00:07:15]Just leave it the way it is.

[00:07:17]Now, we are going to use this to solve our equation quadratic equation using the um we are going to use quadratic equation formula to solve this, which is m equals minus b plus minus we have b squared minus 4 ac all over 2 * a.

[00:07:40]So, what do you have to know?

[00:07:42]Our A is the coefficient of m squared, which is 1.

[00:07:47]B is the coefficient of m, which is 36.

[00:07:51]And C is 36 squared, the constant. So, our m will now be in place of minus B, we write minus 36.

[00:08:00]Plus minus B squared, that's going to be 36 squared.

[00:08:07]36 squared then we have minus 4 multiplied by 1 multiplied by 36 squared.

[00:08:15]Because our our C is 36 squared and A is 1.

[00:08:20]This is all over 2 * 1.

[00:08:23]2 * 1 will still give us 2.

[00:08:26]From here, we have m to be equal to minus 36 plus or minus within this um square root sign, we have a common factor of 36 squared.

[00:08:40]So, let 36 squared come out as a common factor.

[00:08:44]36 squared divided by itself is 1.

[00:08:47]Then, 4 * 1 is 4.

[00:08:50]And 36 squared is already out, so we can close this.

[00:08:55]Remember that everything is over 2.

[00:08:59]Now, we can do something here. Remember that from one of the laws of um sword, if you have the square root of AB, you can express this as square root of A multiplied by the square root of B.

[00:09:16]Okay? So, if this is true, then um I can do what I'm about to do.

[00:09:23]m is equal to minus 36 plus minus we have square root of 36 squared, then multiplied by the square root of 1 minus 4, and that is minus 3.

[00:09:38]This is over 2.

[00:09:40]Now, to go on with this, we're going to have our M to be equal to minus 36 plus or minus square root and square can go.

[00:09:52]And we're going to have 36.

[00:09:55]Then, this negative here is going to give us imaginary, which is I. And that I will multiply by 36. So, we are going to have 36 I. Then, we have root 3.

[00:10:08]Remember, it is this negative that produced the I.

[00:10:12]And everything is over 2.

[00:10:15]But, this is a two-in-one kind of solution.

[00:10:19]Okay? Because of the plus or minus.

[00:10:22]Before anything, let us simplify. Two can go into 36 minus 36 to give us minus 18 plus or minus the same two will go into 36 I, and we're going to have 18 I.

[00:10:39]Then, we have root 3.

[00:10:41]And this is two two-in-one kind of solution.

[00:10:45]So, we're going to bring the three solutions together.

[00:10:50]From our calculation now, the three solutions are M equals 36.

[00:10:57]Then, M 2 equals minus 18 plus 18 I root 3.

[00:11:06]Okay, this is 18 I. Then, our M 3, the third solution, will now be minus 18 minus 18 I root 3.

[00:11:18]Okay, so these are the three solutions.

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