The Unsolvable algebra Equation? Watch this radical trick.

Hi, welcome back to my channel. We have an amazing math question to solve. In this exponential question, we are asked to find the value of x. So looking at the denominator, you can see that both of them are sods, right? And you cannot go further except to rationalize the denominator. So the first thing will be to rationalize. So let's rationalize. To rationalize, we multiply the numerator and the denominator by the conjugate of this denominator. Now, the conjugate of this denominator is 29 plus 1, all right? So we have 14 over 29 minus 1 multiplied by root 29 plus 1 over root 29 plus 1. Now what we have is 14 multiplied by root 29 plus 1 divided by root 29 minus 1 multiplied by root 29 plus 1. So if we move further, we will have 14 brackets root 29 plus 1 divided by, now recall that a squared minus b squared is equal to a minus b multiplied by a plus b, all right? So having said that, that simply means that this will be root 29 squared minus 1 squared. We now have this to be equal to 14 brackets root 29 plus 1 over root 29 squared. This, we have 29 minus 1, all right? So this is equal to 14 brackets root 29 plus 1 multiplied by, divided by 28, sorry. And 14, we go into 28 two times. So it simply means that this 14 over root 29 minus 1 is the same thing as root 29 plus 1 divided by 2, okay? So we're going to keep that aside. We keep that aside and go ahead to rationalize the second, this one. If this is the first time you are seeing this lovely channel, and if you think that you are actually gaining from here, please subscribe to our channel and turn on your notification bell so that you don't miss our amazing videos. We upload our videos as often as possible, so that you're confident to check for our new uploads, okay? Now I'm going to write this here. So this is same thing as root 29 plus 1 divided by 2 raised to the power of x, right? Now let us rationalize this one. Let's rationalize this. So we have 343 divided by square root of 464 minus 1, okay, minus 11, minus 11. So we rationalize by multiplying by root 464 plus 11 over root 464 plus 11. So going further, we have this to be 343 brackets root 464 plus 11 divided by this multiplied by this root 464 minus 11 multiplied by root 464 plus 11. So moving forward, we will have 343 brackets square root of 464 plus 11 divided by, now applying difference of two squares here as well, we are going to have root 464 squared minus 11 squared. So this remove this, we will have 343 brackets square root of 464 plus 11 divided by 464 minus 11 squared is 1, 2, 1. So this will be equal to 343 brackets square root of 464 plus 11 divided by, this will also give us 343, right? So this and this will go and we are left with root 464 plus 11. So therefore, we are now going to replace this with plus, this is root 464 plus 11 raised to the power of 8 is equal to 350. All right, now I want you to notice something. So I'm going to do something right now and I want you to see how this particular action is going to make the whole of this question very easy to solve, okay? So let us take the cube root of root 29 plus 1 divided by 2. Let's take the cube root, okay? We are going to see what will happen. So this will be the same thing as square root of 29 plus 1, everything raised to the power of 3 divided by 2 to the power of 3. So moving forward, remember that a plus b to the power of 3 is equal to a to the power of 3 plus b to the power of 3 plus 3ab squared plus 3a squared b, all right? So that is a plus b to the power of 3. So in this case, a is what root 29 and b is 1. So if we should expand this in this form, we are going to have this to the power of 3. So this is equal to square root of 29 to the power of 3 plus 1 to the power of 3 plus 3 into a is root 29 and b is 1. So 1 squared multiplied by 1 squared, then plus 3 brackets, a is root 29. So root 29 squared multiplied by b, b is 1, okay? So if we expand this, we have square root of root 29 to the power of 3. If we should write this same thing as square root of 29 multiplied by square root of 29 multiplied by square root of 29, right? So you can see that this is root 29 squared. Root 29 multiplied by root 29 will give us 29. So these two is same thing as, or let's take these two. Root 29 multiplied by root 29 is 29. So we have 29 root 29. So we are going to replace this with 29 root 29. So I will now write 29 root 29. So this is equal to 29 root 29 plus 1 plus 3 root 29 because 1 squared is still 1. Then plus, this is 3, let me clean this one so that you don't get confused. Okay. Now plus 3 brackets. Now root 29 squared, this will remove this. So we have 29 multiplied by 1. Now this is equal to 29 root 29 plus 1 plus 3 root 29 plus 3 multiplied by 29. So we have 29 root 29 plus 3 root 29 will give us 32 root 29, 32 root 29 plus 1 plus, now 3 times 29, right? 3 times 9 is 27, right? 7 carry 2. 3 times 2 is 6 plus 2 is 8. So we have 8, 7. So, oh, remember that all these are divided by what? 2 to the 4 of 3. Everything we are analyzing is just this one. The numerator root 29 plus 1 to the 4 of 3. That is what we are still analyzing. So here we will have 32 root 29 plus 88. Okay. So we are going to include this divided by 2 to the 4 of 3. So that means we will now have, so it simply means that what we have now is root 29 plus 1 to the 4 of 3 is equal to this. So we have that's this, which is root 29 plus 1 divided by 2 all to the power of 3. Okay. Will be equal to this, which is this 32 root 29 plus 88 divided by 2 to the power of 3, which is 8. Okay. Now looking at this closely, you will notice something that 8 is a common factor for 32 and 88. So if we factorize it from here, we are going to have this to be 8 bracket 4 root 29 plus 11 divided by 8. Now this 8, we remove this 8. We have 4 root 29 plus 11, right? But we cannot stop here yet. So yes, we cannot stop here. Now let's continue. So moving forward, we will have this to be equal to, if we decide to take this inside, we square it, right? So it will be square root of 29 multiplied by 4 squared is 16 plus 11. Now this is equal to square root of 29 times 16. We give you 4, 6, 4, 4, 6, 4, then plus 11. So you can see that square root of 29 plus 1 divided by 2 all to the power of 3 is equal to square root of 4, 6, 4 plus 11. Okay. So what are we going to do now? So anywhere we see this 4 square root of 4, 6, 4 plus 11, we are going to replace it with this. Okay. So permit me to erase this, this one and replace it with this. Since we are now in agreement that this is equal to this. So because of my board, that's what I'm going to do now. So I will replace this now with square root of 29 plus 1 divided by 2. Okay. Raised to the power of 3, then raised to the power of X. All right. Now we can actually bring this in and take this out without changing the value, right? So permit me to bring this in and take this out. So we have X here and we have 3 here. We have not changed anything. So you can see from here that square root of 29 plus 1 divided by 2 is common between this and this. So we can say, let square root of 29 plus 1 divided by 2 raised to the power of X be equal to Y. All right. So we can call this Y. If we do that, that simply means that that simply means that we will now change this equation to this is what Y. Y plus Y to the power of 3 is equal to 350. Now let's bring this to this side. We will have Y plus Y to the power of 3 minus 350 is equal to zero. So we have minus 350, same thing as minus 343 minus seven, right? So we have Y minus 343 plus Y to the power of 3 minus seven is equal to zero. Now looking at this, we can see that this is the same thing as seven to the power of three. So we have Y minus seven to the power of three plus Y to the power of three minus seven is equal to zero. Now I want this one that is in power of three to be together with minus seven to the power of three. So if I rearrange, I will have Y to the power of three, okay? Minus seven to the power of three, then plus Y minus seven equals zero. Now recall that I want you to remember that I will write this somewhere so that I can clean this. So permit me to write it. Square root of 29 plus one over two raised to the power of X equals Y. Alright, so I can now clean this because I need this space. So moving forward, we have that A to the power of three minus B to the power of three is equal to A minus B, alright? Multiplied by A squared plus AB plus B squared. So it simply means that this becomes Y minus seven multiplied by Y squared plus seven Y plus seven squared, okay? That is it for this. Then plus Y minus seven. So if you look at this, you can see that we have Y minus seven minus seven. Recall that everything here is what is equal to 350. Yeah, but let's just leave it like that or let's write before we forget. So it's equal to zero. Okay, so we continue. We factorize Y minus seven out. We have Y minus seven, open bracket. Then when this is divided by this, this will remain Y squared plus seven Y plus seven squared. That is 49. Then plus one, alright? Because when this is divided by itself is one is equal to zero. So there is no need for this since this is what's positive. Alright, now we have Y minus seven bracket Y squared plus seven Y plus 50 is equal to zero. Now, we're almost at the end of this. Have you subscribed to this channel? If you have not, please try to do that, alright? Yeah, please subscribe to this channel and don't forget to turn on your notification bell. If not, this might be the last time you will see me. Okay, but if you subscribe the algorithm, you have no other choice than to notify you anytime I post a new video. So now we can say that Y minus seven equals zero. So we have Y minus seven is equal to zero or this Y squared plus seven Y plus 50 is equal to zero. So from the left side, we can see that Y is equal to seven. Now, from here, let's check the discriminant because our X is real, not complex. Alright, so we have B squared minus four AC. B is seven minus four multiplied by A is one and C is 50. So by looking at this, you see that we have 49 minus 100, which is going to give us a negative answer. Yes, so if this gives us a negative answer, that means this is rejected because it's going to give us complex roots. So we move ahead with Y is equal to seven. Now, I want us to recall something because that is what will give us the answer that we are actually looking for. Okay, so now recall that whatever I copied the square root of 29 plus one divided by two to the power of X is equal to Y. So now our Y is seven. So we have square root of 29 plus one divided by two raised to the power of X is equal to seven. So at this point, the only way out is to introduce log to both sides. Okay, so if we do that, we are going to have log. So if we do that, we are going to have this will be equal to log square root of 29 plus one divided by two raised to the power of X is equal to log seven. Okay, now remember that when you have something like this, log A raised to the power of B is equal to B log A. So we can write this as X log square root of 29 plus one divided by two is equal to log seven. Now, we are looking for the value of X. Let's divide both sides by this. Okay, so we divide both sides by log square root of 29 plus one over two and both sides. So we divide this side with log square root of 29 plus one over two as well. So you can see that this remove this. Now what we have left is what we have left is X is equal to log seven divided by log square root of 29 plus one divided by two. Now remember that log A over log B is the same thing as log A base B. Okay, using change of base. So that simply means that we are going to have this as X is equal to log seven base square root of 29 plus one over two and that is the value of our X. Thank you so much for watching and I wish to see you in my next video. Bye.