Olympiad Mathematics | Russian | Can You Solve This One?

Added: 2026-05-15

[00:00:01]Hi everyone.

[00:00:05]If you're ready, let's provide the solution to this one right here.

[00:00:13]Okay, we have 8 x to the^ of x = 8 x + 64 65 rather.

[00:00:26]So what do you think we are going to do?

[00:00:31]There's eight over here, right? We can remove this eight by dividing both sides of the equation by 8. So this one will take this out and on the left hand side we'll have x to the^ of x and it's equal to 8 ^ x + 65 all over 8. But you know that the 8 here is the same as 8 to the^ of 1.

[00:01:07]And we have this law of indices that says um if you write a to the power of m over a to the power of n right this can be written as a to the^ m - n right so we apply the same law to what we have there. So x ^ x comes down and is equal to 8 to the^ x + 65.

[00:01:43]Then minus the power from the denominator which is 1. So we subtract 1 from here.

[00:01:53]Now we have x ^ x to be equal to 8 ^ x + 64.

[00:02:03]Now at this point you will see that we have gone a bit far but at this point we still have x spread out. We have x in two places. So our target is to bring the terms with x together.

[00:02:20]And um to do that we'll work on the right hand side there because we know that a ^ m + n is the same as a ^ m * the same a to power of n. So if I apply this to the left to the right hand side then we shall have x ^ x = 8 ^ x * 8 to the power of 64.

[00:02:52]Like I said before our target is to remove or better still.

[00:03:00]Okay. So we still have x spread out. So our target is to bring terms with x together and that will mean that a to the^ x should should be moved to the left hand side and the only way we can do that is to divide this by 8 to the^ x and what you do on one side you have to do on the other side of the equation.

[00:03:26]So this takes this out, right?

[00:03:31]And here we have we're going to have x ^ x over 8 to the^ of x = 60. Okay? Equals 8 to the^ 64, right? So this is what we want. I mean, this is what we have at this point.

[00:03:54]Okay? Okay. So from here now we're going to concentrate on the left hand side first. We are seeing that they both have the same power of x.

[00:04:04]Right? And we know that um m to the power of a / n to the same power of a is the same thing as m / n raised to the same power of a. Okay. So this is very true.

[00:04:23]And um for that we will write what we have on the left in the form of this. So we have x / 8 raised to the same power of x and this is equal to 8 ^ 64.

[00:04:44]Now we look at both sides of the equation.

[00:04:48]It's difficult to make them have the same base. almost impossible but we are going to solve this by comparison. We're going to you know work on the right hand side. So we have x / 8 to the power of x.

[00:05:07]Now this is equal to now look at what I want to do. I want to divide the 64 by 8. So that if I divide 64 by 8, I'll have 8 to the same power of 8. Okay? And to do that, I'll have 6 8 to the^ 64 then raised to the power 1 / 8 because that's the only way I can divide 64 by 8. This 1 / 8 will have to reflect on this side as well. So the whole of this will still be raised to the power of 1 / 8.

[00:05:47]Okay. And we know the relationship between two powers given like this a to the^ m to the^ of n is is equal to a to the^ mn because we multiply the two powers. So that is multiplication and the power becomes okay. So let's get that done. we are going to have x / 8 to the power of x * 1 is x and it's over 8. So on the other side we have six.

[00:06:25]Okay, that is 8. We have 8 ^ 64 thenide by 8 since the relationship between the two powers is multiplication.

[00:06:37]Now we have our x / 8 to the power of x / 8.

[00:06:46]Okay, the same power um the same base, the same exponent, right? So on the other hand, we have 8 to the power of 64 / 8 is 8 and we have succeeded in getting the same base, the same exponent. So at this point we compare the bases or we compare the powers. If you compare the bases x / 8 will be equal to 8.

[00:07:17]Right? And if you still want to compare the powers you still get you still do this and you'll get x / 8 to be equal to 8 as well. So from here now we can easily conclude that we'll get our x from here x * 1 is x and 8 * 8 is 64.

[00:07:41]So from our calculation our value of x is 64. We will not stop here. We have to verify to be sure that we are correct. So let's go into verification phase.

[00:07:56]Okay. So this is the original equation and our value of x is 64 from our calculation. So we'll put in the value of x. We have 8 * x is 64 to the^ of 64.

[00:08:13]Let's see what we have on the right. We have 8 to the power of um um what do we do? x is 64.

[00:08:23]Then we have + 65. So we want to see if the left hand side will be equal to the right hand side. On the left you have base 8 and um this is going to be 8 right * 64 is 8 um squared then multiply by 64 again. So let's see if this will be equal to 8 to the power of 64 + 65 is 129.

[00:08:57]So let's see if the left hand side and the right hand side will be equal. This is 8 * 8 to the^ 2 * 64 is 128.

[00:09:09]And you know that there's an invisible one there, right? And on the right hand we have 8 to the^ 129.

[00:09:16]So from one of the laws of indices which we have talked about pick one of the bases from here we have 8 then add the powers and that will give us 129.

[00:09:27]That is exactly what we have on the other side of the equation.

[00:09:33]So it means that our value of x to be equal to 64 truly satisfies the equation. Thank you for watching.

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