Germany | Can You Solve This? | Math Olympiad

追加: 2026-05-30

[00:00:01]Hello, you're welcome. How do I solve this nice algebra problem? Solution from here.

[00:00:08]From here, we can write this as X over 4 all raised to power 3 equals to here, 27 that's 3 raised to power 3.

[00:00:22]Then, from here this follows when we have A over B all raised to power N is the same thing as A raised to power N over B raised to power N.

[00:00:34]Then, what we have here becomes X raised to power 3 over 4 raised to power 3 equals to 3 raised to power 3.

[00:00:46]Then, we can cross multiply here. This over 1 here.

[00:00:50]X raised to power 3 times 1 still X raised to power 3 equals to 3 raised to power 3 times 4 raised to power 3. We still have that 3 raised to power 3 times 4 raised to power 3.

[00:01:05]Then, also this follows when we have A raised to power N times B raised to power N is the same thing as A times B all raised to power N.

[00:01:18]Then, we can write what we have as X raised to power 3 equals to 3 times 4 all raised to power 3.

[00:01:29]Which is the same thing as X raised to power 3 equals to 12 raised to power 3.

[00:01:38]Then, we take the right hand side to the left hand side.

[00:01:42]And we have X raised to power 3 minus 12 raised to power 3 equals to 0 here.

[00:01:52]Then, this follows when we have A raised to power three minus b raised to power three.

[00:02:01]This same thing as a minus b into bracket open bracket a squared plus ab plus b squared.

[00:02:13]Then this also becomes x minus 12 into brackets and open brackets x squared plus 12 x plus 12 squared close brackets equals to zero here.

[00:02:34]Then we have two possible cases here.

[00:02:37]x minus 12 equals to zero or we have x squared plus 12 x plus 144 equals to zero here.

[00:02:53]Solving on this side, we have x equals to 12.

[00:02:58]Which is a real solution here.

[00:03:01]And here, we have a quadratic equation.

[00:03:03]Here, a equals to one, b equals to 12 and c equals to 1 144.

[00:03:15]Applying the quadratic formula here, which is x equals to minus b plus or minus square root of b squared minus 4 a c over 2 a.

[00:03:29]This becomes x equals to minus 12 plus or minus square root of 12 squared minus 4 times 1 times 144 over 2 times 1.

[00:03:51]Then we have x equals to minus 12 plus or minus square root of 12 squared that's 144 minus 4 * 1 * 144. We can leave it as 4 * 144.

[00:04:08]Then, over 2.

[00:04:12]Then, this becomes x equals to minus 12 plus or minus square root of 144 is common here but 144 into bracket 1 minus 4 here over 2a.

[00:04:31]Next step here, we have x equals to minus 12 plus or minus square root of 144 times 1 minus 4 minus 3 over 2.

[00:04:47]Then, we can separate this one. We have square root of a times b which can be written as root a times root b.

[00:04:55]This here, we have x equals to minus 12 plus or minus root 144 times root minus 3 over 2.

[00:05:09]This we have x equals to minus 12 plus or minus root 144 as 12 and we have root minus 3 as root 3 times minus 1 over 2.

[00:05:24]We can also separate this and we have x equals to minus 12 plus or minus 12 root 3 times root minus 1 over 2.

[00:05:39]And here, root minus 1 is same thing as i.

[00:05:44]It's a complex number here.

[00:05:47]That is we have x equals to -12 plus or minus 12 root 3 i over two.

[00:06:00]That is Here, we can write this x equals to -12 over two plus or minus 12 root 3 i over two.

[00:06:15]Yeah, this becomes x equals to -12 over two, that's -6 plus or minus Also, here we have six root 3 i.

[00:06:27]We have two complex solutions here. Therefore, altogether in this problem, we have three solutions here. One real solution here and two complex solutions here.

[00:06:41]When we write it out, we have x1 equals to 12.

[00:06:45]x2 equals to -6 + 6 root 3 i and x3 equals to -6 -6 root 3 i.

[00:07:03]We have three solutions here in this problem. Let's check if this satisfies this given problem.

[00:07:11]When we substitute the value of x here in this equation given x over 4 all raised to power 3 equals to 27.

[00:07:24]When x is 12 now, we have 12 over 4 all raised to power 3 Is it equals to 27 here?

[00:07:35]Yeah. 12 over 4, that's 3.

[00:07:38]Then raised to power 3 is equals to 27 here.

[00:07:43]And 3 raised to power 3, 3 multiplied itself three times, that's 27.

[00:07:49]Equals to 27 here.

[00:07:52]That is, left hand side now equals to right hand side.

[00:07:58]Hence, x equals to 12 satisfies this given problem.

[00:08:06]When we check the other two complex solutions, which we also satisfy what we have. And thank you for watching. Don't forget these steps.

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