How to solve this Olympiad maths question

Added: 2026-05-18

[00:00:00]Let me show you how to solve this math question on the board.

[00:00:05]They're giving x raised to the power of x equal to 2 raised to the power of 2x + 8.

[00:00:11]And we are expected to find the value of x.

[00:00:16]If you want to solve this question correctly, the first thing you need to do is to interpret what you have on the right-hand side of the equation by using law of indices.

[00:00:28]That's the first thing you need to do.

[00:00:31]So, here's the solution.

[00:00:34]Let us write down what you have on the left-hand side of the equation.

[00:00:38]We have x raised to the power of x equal to Now, if you want to interpret this, this is how to go about it.

[00:00:47]If you look at what [clears throat] you have here, you can see that this two is raised to two powers, one, two, that are being separated.

[00:00:55]The powers are being separated by this um plus sign.

[00:01:00]So, because it is raised to two powers that are being separated, we need to write this number twice because of the two powers.

[00:01:08]So, we have to write it down. This is one, and this is two, because of the two powers.

[00:01:15]Now, after we have done that, carry the first power, which is 2x, and put on top of this first one.

[00:01:22]And then carry the second power, which is this eight, and put on this and put on top of this second one.

[00:01:27]After this, you put multiplication sign in between them.

[00:01:32]The reason we are using multiplication is because the separator of the powers is plus sign, addition sign.

[00:01:39]You understand? In indices, addition and multiplication work together.

[00:01:44]Supposedly, the separator here is subtraction, we would have used division, because subtraction and division work together.

[00:01:51]I hope you understand.

[00:01:53]Now, you need to know that this 2 raised to the power of 2 x, we can simply single out the x.

[00:02:02]Let the x stand alone.

[00:02:04]If we do this, we are not wrong at all.

[00:02:08]It's the same thing.

[00:02:10]Do you understand me?

[00:02:11]>> [snorts] >> Now, let us simplify what we have inside these brackets. We're going to have x raised to the power of x equal to 2 raised to the power of 2 means 2 * 2, which is 4.

[00:02:21]So, we have 4 raised to the power of x times 2 raised to the power of 8.

[00:02:31]Now, the next thing to do is we need to make all the x to be on one side of the equation.

[00:02:38]Here, we have x raised to the power of x. Here, we have 4 raised to the power of x. So, we need to make sure that this 4 raised to the power of x, because it is carrying x, it should move to the left-hand side of the equation so that everything that has to do with x will be on one side of the equation.

[00:02:53]You understand? And for us to do that, we need to divide both sides of the equation by this 4 raised to the power of x.

[00:03:00]So, we divide this by 4 raised to the power of x and also divide this by 4 raised to the power of x.

[00:03:08]On the right-hand side, this 4 raised to the power of x will cancel this one.

[00:03:12]And on the left, we have x raised to the power of x divided by 4 raised to the power of x equal to 2 raised to the power of 8.

[00:03:27]Now, if you look at what we have on the left-hand side, you can see that here we have a fraction.

[00:03:32]And the powers the power of the numerator and that of the denominator are the same. We have x here, we have x here. The powers are the same.

[00:03:40]So, because we have it in this form, we are going to apply something called quotient power law of indices.

[00:03:47]According to quotient power law of indices, anytime in a fraction, the power of the numerator and that of the denominator are the same.

[00:03:54]That we should simply combine the fraction and then raise them to the common power.

[00:03:59]What does it mean? This is what it means. It means that we should write this X over 4.

[00:04:06]And then raise them to the common power.

[00:04:08]The common power here is X.

[00:04:12]Then everything now equals 2 raised to the power of 8.

[00:04:16]>> [clears throat] >> Now, if you look at what you have, you can see that both sides of the equation are in exponential forms, but they do not look alike.

[00:04:28]On the left-hand side, we have a fraction raised to a power, while on the right-hand side, we have just a number raised to a power.

[00:04:35]Not only that, if you look at this left-hand side, you notice that the numerator of this fraction and the power of the fraction are the same.

[00:04:45]Here, we have X at the numerator and the power here is also X.

[00:04:49]While the denominator is 4.

[00:04:52]So, what we need to do is that we need to make this right-hand side to look exactly like what we have on the left-hand side of the equation so that we can compare them and get our answer.

[00:05:01]Now, the question is, how do we make this right-hand side to look exactly like what we have on the left? It's very easy. Let me still show you how to do it.

[00:05:10]So, we have X over 4 in bracket raised to the power of X equal to Now, if you From this now, you can see that this power and this numerator are the same.

[00:05:24]So, that means we need to write this number here, this 2, in form of a fraction such that the numerator is going to be 8 because we have power 8 here.

[00:05:33]We want the numerator and the power to be the same.

[00:05:36]So, this 2 here can be written as 8 / 4 because 8 / 4 give us 2. Right?

[00:05:44]So, in place of this 2, we write 8 / 4.

[00:05:49]And then we raise everything to the power of this 8.

[00:05:55]Now, looking at what we have, you can see that let us compare both sides. You can see that the denominator here is 4.

[00:06:01]The denominator here is also 4.

[00:06:04]Now, let us compare the numerator and the power. Here the numerator and the power are the same. Here the numerator and the power are the same.

[00:06:12]You can see that they look so much alike.

[00:06:15]Now, because they look so much alike, let us compare them. If you compare the powers, you can see that x is equal to 8.

[00:06:21]If you compare the numerator, x is also equal to 8. So, by comparison, we can say in conclusion that x is equal to 8 and this is the answer to this question.

[00:06:33]If you find this video helpful, tell me in the comment section. And please don't forget to share this video so that other students who need this kind of video can also benefit from it.