Olympiad Mathematics | How I got the four solutions

Added: 2026-05-21

[00:00:00]I hope you know how to solve this problem here.

[00:00:05]Okay, if you do not know, then let's solve this together.

[00:00:10]m to the power 4 equals um 4 m to the power of 1 over 3 everything to the power of 3.

[00:00:22]Now, the first thing that will come to the mind um the minds of students is that they will um cancel this 3 and this 3 and that will be completely wrong.

[00:00:36]Yes. Because what we have here, remember that if you have a b to the power of c, this is expressed as a to power c times b to power c.

[00:00:48]Meaning that both 4 and m here are having to do with this power 3.

[00:00:53]So, see how you should simplify that.

[00:00:58]Okay, so you should have m to the power of 4 to be equal to 4 to the power of 3 times m to power 1 over 3 then to the power of 3 as well.

[00:01:13]Right?

[00:01:14]So, from here now we have our m to the power of 4 and it's equal to 4 to the power of 3 then this 3 can go with this 3 and we just multiply by by m.

[00:01:29]Now, [snorts] do not divide both sides by m. What do you do? Bring this to the left-hand side so that we can have m to power 4 minus 4 to power 3 m all equal to 0.

[00:01:45]So, what do we do?

[00:01:47]Remember, how many solutions do you think we are going to have?

[00:01:50]Is it three or four?

[00:01:53]We are going to have four solutions.

[00:01:55]Why? Because the highest power is four.

[00:01:58]So, we're getting four solutions. Now, the first thing you're going to do from here is to factor out your M.

[00:02:06]So, M is a common factor. Here, we have M to the power three remaining.

[00:02:12]>> [snorts] >> Minus here, we have four to the power of three. Remember that this is all equal to zero.

[00:02:19]Yes.

[00:02:21]And [snorts] at this point, we apply our zero product rule to say that M is equal to zero or M to the power of three minus four to the power of three is equal to zero.

[00:02:35]If this is true, then we have a solution already, which is M equals three.

[00:02:42]And like I told you, we are looking for three solutions. Four solutions, right?

[00:02:47]So, we're going to get three more solutions from here.

[00:02:51]What do we do?

[00:02:54]We write our um Let me write it here. M to the power of three minus four to the power of three. This is um equal to zero.

[00:03:06]And it is a difference of two cubes.

[00:03:09]Mind you, A to power three minus B to the power of three is still the difference of two cubes. And it's A minus B into A squared plus AB plus B squared.

[00:03:28]This is our difference of two cubes identity.

[00:03:31]So, our A now is going to be M and our B is going to be four. So, in place of A minus B, we are going to write M minus four.

[00:03:45]And in place of A squared, we are writing M squared.

[00:03:49]AB, that's going to be four * m, and it's 4 m.

[00:03:54]Then we have + b squared. Our b squared is what?

[00:04:00]Our b squared is going to be 4 squared, which is 16.

[00:04:04]Right?

[00:04:05]16. By the way, there's something I would like to do. Instead of writing 16, let me write 4 squared as well.

[00:04:12]Okay, b squared is going to be 4 squared.

[00:04:16]And this is equal to 0.

[00:04:18]Now, I have my reasons for writing 4 squared instead of 16.

[00:04:24]What do we do?

[00:04:25]We apply our zero product rule.

[00:04:28]So, that m - 4 is equal to 0.

[00:04:32]Or m squared + 4 m + 4 squared is equal to 0.

[00:04:41]So, we will come back to this.

[00:04:44]But, before then, from here, we have our m to be 0 + 4.

[00:04:50]And the value of m is equal to 4. We have our second solution.

[00:04:57]Yeah, we're going to bring the solutions together after.

[00:05:00]Now, here we have a quadratic equation, which will give us two more solutions.

[00:05:07]What do I do?

[00:05:08]Let's bring it out, and then we solve it.

[00:05:13]Okay, so this is it.

[00:05:15]The quadratic equation, and we will solve this using the quadratic formula.

[00:05:20]Now, our a is 1, coefficient of m squared.

[00:05:24]And um our b is 4, the coefficient of m.

[00:05:29]And the value of c is 4 squared.

[00:05:33]Okay, I want to use it as 4 squared.

[00:05:36]Now, what is the formula?

[00:05:37]m is equal to - b + - we have b squared - 4 ac This is all over two times A.

[00:05:50]So, what we'll do now is to put in the values of ABC.

[00:05:55]So, that our M will be In place of minus B, we'll put minus four.

[00:06:01]Plus or minus B squared is here as four squared.

[00:06:07]Then minus we have four times one.

[00:06:13]Right? We have four times one. Then we multiply by four squared.

[00:06:20]Because C is four squared from there.

[00:06:23]This is all over two multiplied by one.

[00:06:28]And it will give us one um it will give us two eventually.

[00:06:33]>> [snorts] >> From here, we're going to get M to be minus four plus or minus If you look at this Okay, so from here, we have um four squared.

[00:06:46]And four squared as a common factor. So, four squared will come out.

[00:06:52]And that was the reason I left this as um four squared.

[00:06:56]Then we open brackets.

[00:06:58]Four squared divided by itself is one.

[00:07:01]Four times one is four.

[00:07:03]And this four squared is already out.

[00:07:06]So, we divide this by two times one, which is two.

[00:07:11]So, from here, we are getting our M to be equal to minus four plus or minus we have the square root of four squared multiplied by minus three.

[00:07:26]One minus four.

[00:07:30]And we have this over two.

[00:07:34]Now, you can split what you have under the square root sign.

[00:07:37]So, [snorts] that M will be minus four plus or minus we have the square root of 4 squared multiplied by the square root of -1 um 3.

[00:07:51]And it's all over what? 2.

[00:07:54]So that if we go on from here, we're going to get m to be equal to -4 plus or minus square root of this will go.

[00:08:05]Okay. The square root and the square will go, so we have from 4 then multiply by this negative here is going to come out and we write the square root of -1 as i.

[00:08:17]So what we'll have then now is root 3.

[00:08:20]The negative is no longer there and we are dividing by 2 just like this.

[00:08:28]From here, our m will be -4 plus or minus 2 into Oh, let us divide this already.

[00:08:37]2 into -4 is -2 plus or minus 2 into 4i, that's going to be 2i then we have root 3.

[00:08:47]By the way, this is a two-in-one solution.

[00:08:50]Like I said, we'll bring the four solutions together.

[00:08:54]Before now we got our m1 to be zero.

[00:09:00]Okay, let's write that.

[00:09:03]m1 is zero then our m2 we got it to be four.

[00:09:09]Our second solution.

[00:09:11]Then our third solution is equal to from here we have -2 plus 2i then root 3.

[00:09:22]Then the fourth solution m4 is equal to -2.

[00:09:29]Can you see it?

[00:09:30]Okay, -2 -2i root 3.

[00:09:35]So these are the four solutions to the given equation.

[00:09:41]Thank you for watching.

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