Olympiad Mathematics | Indian | Can You Solve This? | Only the real solutions

Hinzugefügt: 2026-06-01

[00:00:00]If you're ready, let's provide the real solutions for this equation here.

[00:00:07]Okay, solution.

[00:00:10]We have x to the power of x minus the square root of x to the power of 8 equals 0.

[00:00:18]So, how do we solve this problem here?

[00:00:21]The first step, let's take this to the other side. So, we have x to the power x to be equal to the square root of x to the power of 8.

[00:00:34]Now, we will not stop here, right?

[00:00:37]Yes, we're not going to stop here. So, what do we do?

[00:00:40]We will work on this. We know that the square root of a is equal to a to power of 1 over 2.

[00:00:49]Right?

[00:00:50]Okay, so if we know that, then we can write this as x to the power of x being equal to x to the power of 1 over 2, and then this power of 8 is still there.

[00:01:04]And we know the relationship between these two powers.

[00:01:07]Imagine you have a to the power of m, and then you have n over here.

[00:01:13]So, it simply means that you write a, then multiply m by n to get mn.

[00:01:20]So, it's the same thing I'm going to do to the right hand side.

[00:01:25]So, I have x to the power x to be equal to x to the power of You know, we're supposed to have 8 over 2, right? Okay, let me show all the steps.

[00:01:35]So, we have 8 over 2, which is x to the power x being equal to x to the power of 4.

[00:01:43]Okay, so we have x to the power of 4 over there.

[00:01:48]And then if you look at this, let's look at this very well. From here, you should be able to get a solution.

[00:01:56]Why? Because if the bases are equal, if the bases are equal, right?

[00:02:06]If the bases are equal, comma, we equate the powers.

[00:02:19]If the bases are equal, we equate the powers. So, this means that since we have X on the left and X on the right, we can see that X is equal to 4.

[00:02:31]This is coming from the powers.

[00:02:34]This is coming from the powers. But then, I would like us to go back again and look at what we have from here.

[00:02:44]Okay, so let's look at what we have from here. There's a different way we can deal with it.

[00:02:49]Okay, so I'm going to write this now.

[00:02:53]Okay, so X to the power of X is equal to X to the power of 4, right?

[00:02:59]So, from here now, let's divide both sides by X to the power of 4.

[00:03:05]Divide by X to the power of 4, so that this one can go into this.

[00:03:10]And on the left-hand side, X to the power of X over X to the power of 4 is equal to 1.

[00:03:20]Then, on the left-hand side, A to power M divided by A to power N is A to the power of M minus N.

[00:03:32]So, we apply this to the left cuz this is still division. So, we pick one of the bases, then subtract the powers.

[00:03:41]This is still equal to what? 1.

[00:03:45]So, um what do we do from here?

[00:03:48]What do we do from here? Let's do something.

[00:03:51]You know, from what we got, we've already got x to be equal to 4. But, to get that in another way, let's do this.

[00:03:59]We have x to the power of x minus 4 equals x to the power of 0. Okay? I know this will give us 4 as well.

[00:04:08]Because x to the power of 0 will still give us 1. And I wanted to make the bases to be the same.

[00:04:15]So, if the bases are the same, the powers are the same, and we have x minus 4 to be equal to what? 0. And this means that our value of x is equal to 0 plus 4.

[00:04:32]And our value of x finally is 4. This is the solution we got before, right? Now, let's get another solution from here again.

[00:04:43]Okay? Let me take this one off.

[00:04:48]From here on the left, we have x to the power of x minus 4, right?

[00:04:53]And then, remember, if you raise 1 to any number, it is going to be what? 1.

[00:05:00]Yes, even if you have 1 to the power of negative numbers, you're still going to get what? 1. And we are going only after the real solution, so here now, we're going to say that um we have we're going to equate the powers. You know, here we equated the base. Now, we're going to equate the powers. So, I should write 1 >> [snorts] >> Okay?

[00:05:30]I I will write 1 here to the power of x minus 4.

[00:05:36]I will write 1 to the power of x minus 4. In this case, mind you, whatever the value of x is it will not affect the 1 on this side.

[00:05:47]It going to give us one.

[00:05:49]So, if the powers are the same, then we can equate the bases as well.

[00:05:54]Saying that X is also equal to one.

[00:05:58]So, we have two solutions already.

[00:06:01]We have that X is equal to one as our solution, and then we have X to be equal to four. So, these are the two real solutions to the equation.

[00:06:13]But, we have to verify to be sure that both of them satisfy.

[00:06:21]This is the original equation.

[00:06:23]And we have our X to be one.

[00:06:27]Right? And then X to be four.

[00:06:32]So, let's work with one first. We have one to the power of one minus the square root of one to the power of eight. Will this give us zero? Let's try.

[00:06:44]So, one to the power one is one, and the square root of one is also one. So, we have one to the power eight, right?

[00:06:51]And like I said before, one to the power eight will still give us one. So, this is one minus one, which is equal to zero.

[00:07:00]And this means that our X to be equal to one satisfies the equation.

[00:07:07]Now, let's work with the second value, which is four.

[00:07:10]We have four to the power of four minus the square root of four to the power of eight. Will this give zero?

[00:07:18]We have four to the power four minus four to the power of one over two.

[00:07:24]Then we have power of eight.

[00:07:28]Like I explained before, square root is the same as the number or the expression raised to the power of one over two.

[00:07:36]So, we have four to the power four here minus two can go into eight four times. So, that means we are having four to the power of four.

[00:07:45]And we have the same thing. 4 to the power of 4 minus 4 to the power of 4 is zero.

[00:07:50]And this equally means that our X to be equal to four also satisfies the equation.

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