Impossible looking exponential math question.

This looks like an impossible equation, but it's very, very possible. Welcome back to my channel. So today we are asked to find the values of x. Are you ready? Now let's go. The first thing we need to do is this. You can see there's a connection between 3,000 and 2,000. And that is because they share 1,000 as a common factor. So let's bring that out and see how this looks like. So 3,000 is the same thing as 1,000 multiplied by 3. Also, 2,000 is the same thing as 1,000 multiplied by 2. Therefore, we can rewrite this equation as x to the power of 1,000 multiplied by 3 minus x to the power of 1,000 multiplied by 2. And this is equal to 48. Now, let's bring this to this side. We will have x raised to the power of 1000 multiplied by 3 minus x to the power of 1000 multiplied by 2 minus 48 is equal to 0. If this is the first time you have seen my lovely channel, please click the subscription button, turn on your notification bell so that you don't miss my videos. I upload my videos as often as possible. Do tune in at your convenience. There must be something new for you to watch. Give me a thumbs up if you love what I'm doing and most importantly, please drop your comments, the good, the bad, the ugly, anything you feel about my video, let me know, I'll be glad to read that, okay? Recall. a to the power of m times n is the same thing as a to the power of m raised to the power of n okay and also equal to a to the power of n raised to the power of m so these two can be interchanged therefore we can write this as x to the power of one thousand raised to the power of three minus x to the power of one thousand raised to the power of two Minus 48 is equal to 0. Now, you can see x equal to 1000, that's a big number. So, we can actually simplify. So, we can say let x equal to 1000 be equal to a. And if that is the case, that means this equation automatically becomes a to the power of 3 minus a to the power of 2 minus 48 is equal to 0. Alright, now, because I want to factorize this, let's do it this way then. a to the power of 3 minus a to the power of 2, I need a cube from here and a square as well from here. And the available cubes and squares is going to be negative 64 and plus 16, right? Because this plus this will give me negative 48, is equal to 0. Now, 64 is 4 to the power of 3, so it's a cube, and 16 is 4 to the power of 2. So, we can write this as a to the power of 3 minus 4 to the power of 3 minus 64 is 4 to the power of 3, and plus 16 is 4 squared is equal to 0. Now, let's collect like terms. So we have a to the power of 3 minus 4 to the power of 3. Then we have minus a squared and plus 4 squared is equal to 0. Now, recall, I want to put this in brackets. So I can write this as a to the power of 3 minus 4 to the power of 3. Then minus. This becomes a squared. But this becomes minus 4 squared is equal to 0 because minus minus will give me plus. So now remember that a to the power of 3 minus b to the power of 3 is equal to a minus b multiplied by a squared plus ab plus b squared. And also a squared minus b squared is equal to a plus b multiplied by a minus b. So I'm going to expand this equation now. Using these two algebra identities. Have you subscribed to my channel yet? If not, I think this is the right time to do that, okay? Alright, now we have... This will now be written in this form as a minus 4 multiplied by a squared plus 4a plus 4 squared, right? That is it for this one. Then minus, that means I need to arrange this side as well, okay? So minus, for this, we are going to have a plus 4 multiplied by a minus 4 is equal to 0. Now, I want us to look closely because you are going to notice that we have a minus 4 here and a minus 4 here. So, we can actually factorize that out. So, if we do that, we have a minus 4, open bracket, this one. So we're going to have a squared plus 4a plus 4 squared and then we are going to have minus bracket a plus 4 for this one and this will be equal to 0. So if we continue, we are going to have a minus 4 open bracket a squared plus 4a plus 4 squared. Now this minus, we open this bracket as minus a and minus 4. This is equal to zero. Now let me come to this side. So moving forward we have a minus four bracket a squared plus 4a minus 4 a is plus 3a and 4 squared is sixteen. Sixteen minus four is twelve so we have plus twelve is zero. Now from here you can see that a minus four is zero And a squared plus 3a plus twelve is also zero Remember, we are not looking for a, we are actually looking for the values of x, right? But from here, if this crosses, we have that a is equal to 4. Then from here, let's go ahead and solve. From here, we are going to use the quadratic equation formula that says that a will be equal to minus b okay let's use x okay x is equal to minus b plus or minus square root of b squared minus 4ac everything divided by 2a now in this case a is what 1 b is 3 and c is 12. i'm talking about a b c of this particular formula So, we can now solve this x. Remember, this x is representing what we are looking for here, which is this x. So, this x now will be equal to minus 3 plus or minus square root of 3 squared minus 4 multiplied by 1 multiplied by, our c is 12, divided by 2 multiplied by So our x will be... Minus 3, plus or minus. The square root of 9 minus 48 divided by 2. Now, from our discriminant, we are going to notice that we are going to have a negative here, which means that there is no real root, right? No real roots. So we are going to what? We are going to discard this one. We are going to leave this one. So we are concentrating on a equals 4 right now. So, we are going to find the values of our x using a equals 4. So, but a is equal to 4. And recall that we have that x to the power of 1,000 is equal to a, right? Therefore, we can say x to the power of 1,000 is equal to 4. Now, let's raise both sides of this equation by 1 over 1,000. So we have x to the power of 1,000 raised to the power of 1 over 1,000 is equal to 4 raised to the power of 1 over 1,000. Now on this and this we go, we have x is equal to 4 raised to the power of 1 over 1,000. But we can still express 4 as 2 to the power of 2, right? So we have x equals 2 to the power of 2 raised to the power of 1 over 1,000. Which we can divide this and this to have 500, okay? So finally, we can now say that x is equal to 2 to the power of 1 over 500. Thank you so much for watching and see you in my next video. Bye.