Germany | Can you solve? | A nice maths olympiad question |

追加: 2026-05-20

[00:00:00]Hello everyone. Welcome to Rasta's classroom. Today we are solve a interesting viral math olympiad question.

[00:00:08]So how to solve this German math olympiad question which is x + 8 whole to the power 6 is equal to 64. x is equal to what? How to solve this math problem?

[00:00:20]So I solve this question is a method.

[00:00:23]It is a long math problem.

[00:00:26]Our math solution.

[00:00:30]First of all, I take our question which is x + 8 bracket whole to the power 6 is equal to 64.

[00:00:42]Now I move on this 64 in left side which is x + 8 bracket power is 6 - 64 is equal to 0.

[00:00:56]Then this 6 equals to 0 + 8 bracket power is 3 bracket power is 2.

[00:01:11]At this moment you can say this 2 times this 3 will be 6.

[00:01:18]Because of that I want to apply here is math formula.

[00:01:21]Then minus 64.

[00:01:23]So we know that 64 it will be 8 to the power 2 is equal to 0.

[00:01:28]8 * 8 this is 64.

[00:01:32]Now you can apply here is math formula a square minus b square. So it will be a to the power 2 - b to the power 2 which is a + b times a minus b.

[00:01:48]I apply this math formula here.

[00:01:51]Now this expression it will be 8 x + 8 bracket power is three plus eight a plus b times a minus b a is x plus eight bracket power is three minus b b is eight is equal to zero Now at this moment we are find out two case so this product will be x plus eight bracket power is three plus eight is equal to zero and we are find out others case which is x plus eight bracket power is three minus eight is equal to zero so we are find out here is two case now at this moment this expression equals to this is x plus eight bracket power is three and this eight it will be two to the power three two to the power three this is eight now you can apply here is a cube plus b cube formula which is a cube plus b cube we know that it will be a plus b times a square minus a b plus b square I apply this math formula here so this expression it will be x plus eight plus two here is a is equal to x plus eight b is equal to two then here is a square so it will be x plus eight bracket power is two minus a b a is x plus eight [clears throat] and b is 2 + b squared. I mean 2 to the power 2 is equal to 0. I take this case. Later we are take this case. Okay.

[00:04:04]Now, this is x + 8 + 2, this is 10.

[00:04:09]times and this is a + b whole squared, so which is x squared + 2ab + b squared.

[00:04:16]So, it will be x to the power 2 2ab 2 * 8 16 16x + b squared. I mean 8 to the power 2, this is 64 - 2 * x 2x then + - this is - 8 * 2, this is 16 and 2 to the power 2, this is 4 is equal to 0.

[00:04:40]Then this expression it will be x + 10 times and this is x to the power 2 Now, this is 16x, this is 2x. So, if I take the 16 and 2x, it will be 14x.

[00:05:00]We are find out here is 14x.

[00:05:04]Then this is 64. 64 - 16, it will be 48.

[00:05:09]48 + 4, this is 52.

[00:05:12]So, we are find out here a nice quadratic equation here.

[00:05:17]So, we are find out two cases. Our first case, which is x + 10 is equal to 0.

[00:05:23]And other case, which is x to the power 2 + 14x + 52 is equal to 0.

[00:05:33]Now, at this moment you can easily hear our real solution x is equal to -10.

[00:05:40]So, we are find out x is equal to -10.

[00:05:42]But here, x is equal to what?

[00:05:45]So, at this moment you can say it will be x is equal to if I apply this quadratic formula, - b plus minus square root b squared minus 4ac over 2a.

[00:06:01]Then, this expression, it will be x is equal to minus b.

[00:06:06]B is equal to 14. So, I take this 14 here, plus minus square root b squared. So, it will be 14 squared minus 4ac. 4a, a is 1 and c is 52 over 2 * a. This is a is 1.

[00:06:30]Then, this expression, I can say it will be x is equal to minus 14 plus minus square root.

[00:06:40]Then, 14 squared, it will be 196 minus 4 * 52. It will be 208 over 2.

[00:06:51]Now, this is minus 14 plus minus square root.

[00:06:58]Minus 12.

[00:07:00]This minus this, it will be 12 over 2.

[00:07:03]Okay, solve this interesting math problem step by step. X is equal to minus 14 plus minus 4 times 3 times negative 1, it will be 12 negative 12 over 2. Then, this is x is equal to this minus 14 divide 2, it will be minus 7.

[00:07:27]I take this minus 14 here, okay? Then, plus minus 2 square root 3. And a square root negative 1, it will be i. It is a complex number over 2.

[00:07:38]Now, you can say here, x is equal to this 14 divide two, it will be minus seven.

[00:07:46]Plus or minus this two this two cancel, it will be square root three i. So, we are find out here is two solution, both are complex number, which is minus seven plus minus square root three i. And here is real solution x is equal to minus 10.

[00:08:03]Now, at this moment we are solve others case uh which is x plus eight minus eight equal to zero.

[00:08:13]I solve this case at this moment. This is our first case. This is our second case. And we are find out here is six root. We are find out three root. At this moment we are find out here is three root.

[00:08:25]So, let's start. Our equation which is x plus eight bracket power is three and this eight it will be two to the power three is equal to zero.

[00:08:39]Now, I apply here is a cube minus b cube formula.

[00:08:44]So, it will be a minus b times a square plus ab plus b square.

[00:08:54]Now, this expression it will be a plus b a minus b, so it will be x plus eight minus b two then this is a square, it will be x plus eight bracket power is two plus ab x plus eight times b is two plus b b square. So, b it will be two to the power two is equal to zero.

[00:09:23]Now, this expression it will be x plus six.

[00:09:27]This is eight, this is two, it will be six.

[00:09:30]And this is x square. I apply this formula a plus b whole cube as a square it will be a square plus two ab plus b square. So, it will be 16 x plus 80 square. This is 64. And this is twice x and this is 16 two times eight. It will be 16.

[00:09:48]This is 16 and this is two to the power two is equal to four is equal to zero.

[00:09:57]Now, this expression it will be x plus six times and this is x to the power two.

[00:10:05]And this 16 and this two it will be 18 x.

[00:10:12]And plus this is 64.

[00:10:16]And this is 16 and this is four.

[00:10:19]So, it will be 84 is equal to zero. So, we have find out here is two case. Our first case at this moment x plus six is equal to zero.

[00:10:33]And others case which is a quadratic equation square plus 18 x plus 84 is equal to zero.

[00:10:42]Now, at this moment you can say here x plus six which is x is equal to minus six. This is our real solution, but here is x is equal to what? So, if I apply this quadratic formula here x is equal to minus b plus minus square root b square minus four a c over two a.

[00:11:08]If I apply this master formula here, so this expression it will be x is equal to minus b.

[00:11:15]B is equal to minus 18 plus minus square root 18 square minus four a c over two a. A is one.

[00:11:30]Now, this is x is equal to minus 18 plus minus square root 18 square. So, it will be 324.

[00:11:41]18 * 18 minus 84 * 4. It will be 336 over two. 2 * 1, this is two.

[00:11:52]Now, at this moment [clears throat] you can say here X is equal to minus 18 plus minus square root negative um this minus this it will be 12 over two. So, it will be minus 18 plus minus square root 12, which is a square root 4 * 3 * minus 1 over two.

[00:12:16]Now, this is X. X is equal to minus 18 plus minus square root 4, which is two, and this is a square root 3, and a square root negative 1 it will be I.

[00:12:27]This is complex number over two. Now, we're find out this 18 / 2 minus 9 plus minus this two two cancel it will be square root 3 I. So, we're find out here is complex solution. So, our final answer which is our sixth answer, which is X1 is equal to 10 negative 10.

[00:12:52]Uh we're find out here X1 is equal to This is minus 10, yes.

[00:13:04]And we're find out uh here X2 is equal to minus 7 plus minus square root 3 I.

[00:13:19]This is uh three solution. It will be two three.

[00:13:23]And we're find out here is three solution also. X is equal to minus six.

[00:13:28]So, X is equal to minus six.

[00:13:31]And we'll find out other solution, minus nine plus minus square root three I.

[00:13:38]So, we have find out here is six solution, here is three solution, here is three solution. This is our final answer in this tricky math problem.

[00:13:45]Let's check out or let's verify our question, which is X plus eight whole to the power six is equal to 64.

[00:13:57]This is our question. I take our real solution, X is equal to negative 10. So, negative 10 plus eight bracket power is six is equal to 64. Is it right? Yes, right. Because of that minus two power is six, this is always 64. This is positive because of that minus something power is even number, it is always positive. Now, I take minus six. So, minus six plus eight bracket power is six, which is two to the power six. Two to the power six it will be 64. So, left hand side and right hand side both side is equal. So, this is our final answer in this tricky math problem we had question.

[00:14:39]Thank you, everyone. Have a good day.