INTERNATIONAL MATHEMATICS OLYMPIAD | FRANCE|

[00:00:00]Most students cannot solve this amazing Olympian math question on the board. But in this video, I'm going to reveal everything that you need to know about this equation. Watch this.

[00:00:10]We have x - 1 to the power of x equals 2 to the power of 10, and we are asked to find the value of x. Can you solve this? Anyways, let me help you out.

[00:00:20]Now, look at this thing.

[00:00:22]We don't have any business with the left-hand side of the equation.

[00:00:25]So, let us tackle this 2 to the power of 10 to make sure that we express this to behave exactly like this.

[00:00:32]So, now we shall have x - 1 all to the power of x Okay?

[00:00:40]To be equal to 2.

[00:00:42]Now, we can write this as 2 * 5.

[00:00:49]All right?

[00:00:51]We can write this in this form.

[00:00:53]Because 2 * 5 is 10.

[00:00:56]So, we can first simplify this as x - 1 all to the power of x is equal to We can also rewrite this as 2 to the power of 2 all to the power of this 5.

[00:01:17]Okay? Because from this law of indices, if you have a to the power of x all to the power of y we can equally write this as a to the power of x y. So, if you single out or if you remove these brackets, you can multiply this, which result to this, and 2 * 5 will be equal to 10. So, we get the original equation.

[00:01:44]All right. So, what next?

[00:01:47]What is 2 to the power of 2? 2 to the power of 2 is 4. So, the equation becomes x - 1 all to the power of x to be equal to um four to the power of five.

[00:02:03]But, look at the equation very well. You cannot compare because this is x - 1 and this is also four.

[00:02:12]What is this x and this is five.

[00:02:16]But, how can you solve this?

[00:02:17]Make sure that remember our first statement we said that I want to make the left hand side or the right hand side to be like the left hand side. Of course, we don't have any business with this. So, we try to express this four to have something like this.

[00:02:32]So, now of course, we know that four can be expressed as five minus one, right? So, we shall have our x minus one all to the power of x to be equal to instead of this four, impute your five minus one because five minus one is equal to four, right? All of this to the power of five.

[00:02:55]What next?

[00:02:57]Now, let us deal with the base first before we can equate the powers. Let's see whether the base will correspond with the solution to the what? Exponent or the power. So, firstly, we can say x minus one equals five minus one. Of course, this is a linear equation. It's very easy to solve.

[00:03:19]So, we shall have x minus one to be equal to five minus one is four.

[00:03:25]So, we can say x here is equal to four.

[00:03:29]When minus one crosses over, we shall have plus one, right?

[00:03:32]So, our x here will be equal to four plus one is five.

[00:03:37]So, from the base, we see that x is equal to five. But, from the exponent, you don't need to waste much of your time. Just notice that x is what? x is equal to five, which means x as five satisfy this equation. If this video is interesting, effort to share this video, follow us, and subscribe to this channel for more math tips like this. Thank you.