Olympiad Mathematics | How I got the three solutions | Germany

Added: 2026-05-12

[00:00:02]Okay, if you're ready, let's solve this one very quickly.

[00:00:11]We have the square root of m to the^ of 6 to be equal to 8 to the^ of -1.

[00:00:22]Okay. So from the left hand side we can simplify what we have there because roo<unk> of m ^ 6 can be written as m to the power of 6 to the power of 1 / 2. Okay. This is the same as the square root of m to the power of 6.

[00:00:46]And this is equal to from here we have um 8 to the power of -1.

[00:00:55]So um what do we do from here?

[00:00:59]We go over to the left hand side. These two can go into the six and we'll have three. So m to ^ 3 like this is equal to by the way we have 1 / 8.

[00:01:19]So how do we now solve this problem here? You can multiply you know both sides of the equation by 8 right. So we have 8 * m cube and is equal to 8 * 1 / 8. So that this way we can easily cancel out 8 from the right and we have 8 m cub 8 m cub to be equal to 1. Now see what I want to do um I have um 8 mq but 8 can be written as 2 ^ 3 and we multiply by m^ 3 and this is equal to 1.

[00:02:04]But then from one of the laws of indices we know that we can multiply 2 by m to get 2 m. Then both of them are going to have the same power of 3.

[00:02:19]So that everything here is equal to one.

[00:02:23]Now we are even interested in you know equating the powers right. So to equate the powers we can write this as one 1 to the^ 3 because 1 to^ 3 will still give us 1. So let's bring everything to the left hand side and we'll have 2 m to the power 3 - 1 ^ 3 = 0.

[00:02:49]So that from here we can apply our popular difference of two squares I mean difference of two cubes and it says that a cub - b cube is a - b * a 2 + a b + b 2 right so this is what it is and from here now we will just put in the values because our a in this case is going to be 2 m and our b is going to be 1. So we're going to put in the values right about now. So in place of a minus b we write 2 m - 2 m -1 in bracket then open bracket a² is going to be 2 m. Okay, I have to change this bracket so that we can square this one as well.

[00:03:51]Then we have plus a here is 2 m. Then multiply by b which is 1. Then we have + 1 2 and we equate all of these to zero.

[00:04:05]Sorry, I wrote out of sight.

[00:04:08]Right, let me place it very well. So this is what we are talking about. Now the next thing we're going to do is to work on the middle um the terms in the second bracket. Right?

[00:04:21]Okay. We're going to work on the terms in the second bracket. Let's do that right away.

[00:04:27]We will now try to simplify the terms here. But we know that 2 m - 1 is in the first bracket.

[00:04:37]Then here 2 m ^ 2 is going to be 4 m 2.

[00:04:42]Then + 2 m * 1 is 2 m. Then we have + 1 2 which is the same as 1. This is all equal to zero. So we apply our zero product rule. Since we are multiplying these two terms to get zero. So either of them must be zero. Picking this one first. we have 2 m -1 to be equal to zero. Meaning that 2 m is = 1, right? If we take one to the other side and then our m will just be 1 / 2. By the way, this is a solution on its own.

[00:05:27]Yeah. So to get the other solutions, we are going to pick that quadratic expression there and equate it to zero.

[00:05:35]If we do that, it becomes quadratic equation. 4 m 2 + 2 m + 1 = z. This is our quadratic equation. And we can get our a b c. a is a coefficient of um m² that's going to be 4. And b is a coefficient of two of m which is going to be two. And our c is a constant. So C is equal to 1 the constant. The next you do is to get your quadratic equation formula and it says that M is equal to - B + - we have B ^ 2 - 4 A C all over 2 * A.

[00:06:27]So that we can now put in the values of a bc that we have got and our m will be equal to place of minus b we're going to write -2 right then plus or minus we have b² our b square is going to be 2^ 2 then we have - um we have 4 * a it's also four then multip multiply by C. Our C is 1. Now this is all over 2 * 4. Yeah. 2x4 because we have our 2x4 because we have our a to be 4. So from here we have to simplify further and our m will be -2 + - 2 ^ 2 is 4 then 4 * 4 is 16 divide this by 2 * okay 2 * 4 right and that is 8 so we put 8 directly so to go on we have m to be = -2 2 and we have plus or minus the square root of -12 because 4 - 16 is -2 and we have this over over um 8. By the way we having negative root right? So we can simplify that to get - 2 + or minus we get I<unk>2.

[00:08:10]What I have done is to pick out the negative there. So the negative has come out and we have all [snorts] of this over it. Now let's simplify further.

[00:08:22]Okay. So to simplify this further we shall have m to be = -2 [snorts] + or minus we have i and our square root of 4 [snorts] square root of 12 is 4 * 3.

[00:08:37]Okay. So this is over over 8. Now let's go on. You'll see how interesting this is. Our M is going to be - 2 plus or minus we [snorts] have I multiply by the square root of 4 is 2. Then multiply by the square root of 3 because 3 is not perfect square. It has to come down.

[00:09:02]This is 8.

[00:09:04]Okay. Now we have our m to be equal to -2 + - i * 2 is 2 i and we have <unk>3 and all of this is over 8. So how best can we simplify this? We're going to have m to be equal to from the numerator 2 is a common factor. So bring out your two open bracket. Two out of that is -1 + minus 2 out of that is i. Then we have <unk>3 and all of this is over eight. Right?

[00:09:40]So we can break it. 2 into itself is 1.

[00:09:43]2 into 8 is 4. So at the end of the day we have m to be = -1 + or minus i <unk>3 all over 4. So we have this point our two in one solution.

[00:10:01]Now we're going to bring the three solutions together at this point and we have m to be equal to 1 /2.

[00:10:10]This was our first solution. Call it m1.

[00:10:14]Then our second m_sub_2 is going to come out of this. We have -1 + i <unk>3 / 4. This our second solution. Then our third m3 is -1 - i <unk>3 all over 4. By now we have the three solutions to this equation. Thank you for watching and I believe you've um got one or two things new from this video. Thank you for watching and always you know share to your friends that you think would need this. Thank you for watching. Subscribe.