Very tricky exponential math equation. Is the answer really 4!

Some people will say that x to the power 6 to the power of 6 equals 4 to the power of 6. This implies that x equals 4. That's not true. Okay? Now, I'm going to give you a step-by-step approach. on how to find all values of x. We are supposed to get 6 values of x. Are you ready? Now, let's go. The first thing we need to do in order to avoid this mistake here to raise 4 to the power of 6 to the other side of the equation. So, we are going to have x to the power 6 minus 4 to the power of 6 equals 0. Alright? Now, the next thing is this. Note that 6 is equal to 2 multiplied by 3. Therefore, we can express this as x cubed multiplied by 2. Minus 4 to the power of 3 multiplied divided by 2 is also equal to 0. Now remember once again using the law of exponents that a to the power of m times n is equal to a to the power of m times n of m mod m raised to the power of n, right? a to the power of n raised to the power of m Both of them can be interchanged. So, using that law, we can actually express this as x to the power 2 raised to the power of 3 minus 4 2 raised to the power of 3. Or, we can write this as x cubed raised to the power of 2 minus 4 to the power of 3 raised to the power of 2. and we are going to use this one because this is giving us the difference of two squares, right therefore we are going to rewrite this as x to the power of three squared minus four to the power three squared is equal to zero, all right But before I continue, have you subscribed to my channel? if this is the first time you've seen This lovely channel, please hit that subscribe button and turn on the notification bell. Your notification bell is very important so you don't miss our videos, all right? We upload our watch videos as often as possible At your convenience, there must be something new for you. Watch now, give us a thumbs up if you love what you're doing and please always drop your comments Very important to me, I always like to read the good The bad, the ugly—just tell me what I'm doing. You like it, and where do I need improvement? Okay, thank you. Thank you very much. Now let's continue, so from here. Using the difference of two squares, that says that a squared minus b squared is equal a plus b multiplied by a minus b, right? Therefore, we can write this as x to the power three plus four to the power of three, right? Now, I will define a value x to the power 3 plus 4 to the power of 3 multiplied by x cubed minus four cubed and this is equal to zero Now, we can actually express this as x to the power 3 plus 4 to the power of 3 is equal to 0. is equal to 0. Now, recall the algebraic identity that states that a cubed plus b cubed is equal to a plus b multiplied by a squared minus ab plus b squared Alright? And another one that says a cubed minus b cubed is equal to a minus b multiplied by a squared plus a squared plus 2ab plus b squared. Now, using these identities, we are I am going to expand this particular equation, this one. and this one, so for this we have x to the power of Three plus four to the power of three equals zero. and on this side we have x to the power of three minus four to the power of three equals zero so for this we are going to have x I need a little space. So for this, we are going to have x plus 4 multiplied by x squared. minus 4x plus 4 squared, right? This is equal to zero Okay is equal to zero, and for this we are going to have x minus 4 multiplied by x squared plus 4x plus 4 squared equals zero So from here we now have 2 equations. From here and two equations from here as well so from here we have x plus 4 equals 0 or x squared minus 4x plus now 4 squared is 16 is also equal to 0, right? That is for this side. Now, on the other hand, We have x minus 4 is equal to 0, or x squared plus 4x plus 16 equals 0. Now, I'm going to erase this side of the board. And try to solve this one after the other. Okay, so from here, we can see that x plus 4 equals 0. Now, if this crosses, we have that x is equal to negative 4. And this becomes the first value of our x. Remember, we were expecting to get 6 values of x, right? And this is the first value. Now, for this, let's do the same here. So, now we can obtain the second value. So, we have x equals, this cross becomes plus 4. Thus, this becomes the second value of x. So, we still have 4 more values of x to obtain. So, with that, this is what remains to solve this one And this one, and I'm going to copy them out. So, if I copy this one, I will have x squared minus 4x plus 16 equals 0, and I'm going to erase this, and then we have x squared plus 4x plus 16 is also equal to 0. So, we have obtained x1 and x2 as plus 4 and minus 4. Let me write it down somewhere, plus 4 and minus 4. Alright, now, moving forward. From here, we are going to solve using the quadratic formula. Alright, now, using the quadratic formula that states that x is equal to minus b plus or minus the square root of b squared minus 4ac, all over 2a. Using this particular formula, in the first equation, We have that our a is 1, the coefficient of x squared, our b is negative 4, the coefficient of x And our c is 16, which is our constant. So moving forward, we substitute. So, for this one, x will be equal to minus minus 4 plus or minus square root negative 4 squared minus 4 multiplied by our a is 1 and our c is 16, everything divided by 2 multiplied by 1. Similarly, this side will be x equals to minus 4 plus or minus square root 4 squared minus 4 multiplied by 1 multiplied 16 over 2 multiplied by 1. Now, looking at this, we have that x is equal minus times minus is plus, so we have plus 4, plus or minus, square root of minus 4 squared is 16, minus, Now our b is, oh, okay, 4 is here, this is correct, so 4 times 16, So we have 4 multiplied by 16, all over 2. So our x will be 4 plus or minus the square root of here, We can factor out 16, open bracket. Now when 16 is divided by 16, we have 1 minus, When this is divided by 16, we get 4, all over 2. Likewise, x equals minus 4 plus or minus square root 16 minus 4 multiplied by 16 divided by 2. So, we have x is minus 4, plus or minus, square root of, here we also factorize, 16 bracket. When 16 is divided by itself, we have 1 minus. When this is divided by 16, 16 goes in 4 times with a remainder of 4. Now, everything divided by 2. So, coming to this side, we have x is equal to 4 plus plus or minus the square root of 16 multiplied by negative 3 right, divided by 2. And here, we have x equals negative 4 plus or minus the square root of 16 multiplied by negative 3 as well, all over 2. Now, looking at this, this will be equal to, because I have limited space, I'm trying to see if I can finish everything on one side of the board. So, this will be, x will be equal to 4 plus or minus, Now, the square root of 16 is 4, right? Then, we have the square root of negative 3, everything divided by 2. and this is the same x equals negative 4 plus or minus the square root of 16. Now recall that the square root of a b is the same as the square root of a multiplied by the square root of b that is why I am separating this as the square root of 16 multiplied the square root of negative 3, okay, and the square root of 16 is 4 Then we're okay, sorry, my camera stopped recording, I didn't even know. And this is where it stopped recording right here. For this side, we have this. and remember that the square root of negative 1 equals i, and we can express this as x equals 4 plus or minus 4 times the square root of 3 multiplied by negative 1,报名 divided by 2. So, from here we have x equals minus 4 plus or minus square root of 16 multiplied by square root 3 times the square root of negative 1, right? Everything divided by 2. So, here we have x equals 4 plus or minus 4 root 3 i, okay, divided by 2. And here x equals negative 4 plus or minus 4 times the square root of 3i divided by 2. So here we have x equals, This we divide to give us 2 plus or minus. This will also give us another 2 plus 2 root 3i. And here we have negative 2 plus or minus. This will go into this 2 root 3i. So remember, we have obtained x1 and x2; x1 is 4, x2 is negative 4 now here we have x3 and x4 2 plus or minus And here we have x5, x6, and these are the 6 values of x. Thank you so much for watching, thank you for always commenting. Thank you for always advising me; I really appreciate you so very much. Yes, you, in particular, who always watch my video to the end and comment and tell me how you Feel free, thank you so much for your support And I wish to see you in my next video. Bye.