Olympiad Mathematics | Germany | Can You Solve This?

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[00:00:00]Hi, everyone.

[00:00:04]You're welcome.

[00:00:06]Let's provide the solution to this equation here.

[00:00:12]Okay, so we have x to the power of 3 plus x to the power of 1 over 2 equals 66.

[00:00:20]Now, how do you solve this problem?

[00:00:24]Now, let me pick out what we have here, right?

[00:00:28]Now, we have x to be what? x to the power of 1 over 2.

[00:00:33]Let's this be equal to y.

[00:00:37]Right?

[00:00:38]So, that anywhere we find x to the power of 1 over 2, we're going to write y.

[00:00:43]Then, how do we do this? What do we do to this one here?

[00:00:49]Now, from here we have to remove the power of 1 over 2 by squaring it, right?

[00:00:58]And on the other hand, we're going to square the y.

[00:01:02]This will take this out.

[00:01:05]And at the end of the day x will be equal to y squared, right?

[00:01:12]Okay. So, if this is true then we can put these values you know, this y and y squared into the equation.

[00:01:23]Permit me to write the equation again so that we can see it here.

[00:01:27]x to the power of 3 plus x to the power of 1 over 2 equals 66.

[00:01:35]Now, this is x to the power of 3, but we now know that our x is y squared, right?

[00:01:42]So, that means that in place of x cubed I'm going to write y squared.

[00:01:48]Then, there's power of 3.

[00:01:51]Plus, look at that.

[00:01:53]Um y to the power x to the power of 1 over 2 is Y from there.

[00:01:58]So, this is Y and it is all equal to 66.

[00:02:04]We will not stop here, right?

[00:02:07]So, to continue with this, we know the relationship between these two powers because A to the power of M to the power of N can be written as A to the power MN.

[00:02:21]Right? Because what we have done is just to multiply by M by N.

[00:02:29]Okay, let me remove this for the sake of space.

[00:02:33]Now, we are going to have Y to the power of 6 plus Y equals 66.

[00:02:41]Now, we have to look at what we have on the right because definitely we cannot add these two.

[00:02:50]But then, from here, let's bring out a number from 66 that can be expressed to have the power of 6.

[00:03:00]Okay, I already have a number.

[00:03:02]Remember, you can bring out 40, you can bring out 60, but you cannot express them to the power of 6.

[00:03:11]So, what do we do?

[00:03:13]For Okay, let me write this first. Y's Y's Y to the power of 6 plus Y equals 64 plus two.

[00:03:24]64 plus two. Oh, by the way, we are getting only the complete solution I mean the real solution.

[00:03:31]We are getting only the real solution.

[00:03:34]So, we will not have to go too far.

[00:03:38]Okay. So, from here now we have Y Y to the power of 6 plus Y is equal to look at 64.

[00:03:51]To express 64 in the power of six, you're going to write two to the power of six.

[00:03:58]Then we have plus two.

[00:04:01]So, at this point, we can begin to compare.

[00:04:07]Okay, so by comparison, we're going to say um y to the power of six should be equal to two to the power of six.

[00:04:19]And now that the powers are the same, we can equate the bases.

[00:04:24]So, that y will be equal to two.

[00:04:28]Right?

[00:04:29]And then or if you compare this and this, you're going to get y to be equal to two directly.

[00:04:40]But we will not stop there, right?

[00:04:42]Because we are getting the value of x.

[00:04:46]And at some point, we said that let x to the power of 1 over 2 be equal to y.

[00:04:56]Okay?

[00:04:57]So, if this is true, then we can say that You know, the value of y is two now, so we can say x to the power of 1 over 2 is equal to two.

[00:05:08]And to remove that power from there, we will have to square both sides of the equation.

[00:05:16]And then x will be equal to two squared, which is Right? So, this right now becomes the value of x that will satisfy the given equation, which is x to the power of three plus um we have um x to the power of 1 over 2 equals 66.

[00:05:42]This is the equation, right? We want to verify.

[00:05:45]If you put in the value of x to be four, then you have four cubed plus four to the power of one over two.

[00:05:56]Four cubed is 64.

[00:05:58]Then four to the power of one over two is is two.

[00:06:02]Cuz this is the same as the square root of four.

[00:06:05]So at the end of the day we have six plus four, which will give us 66.

[00:06:11]So the value of x to be equal to four truly satisfies the equation.

[00:06:17]But mind you, this is only the real solution to the equation.