A Nice algebra simplification Math problem| can you solve for x?

Subscribers, thank you for joining us today. If you're new, and if you're a returning viewer, Thank you so much, I appreciate it. Now today we are asked to find the value of x. Let's restart from Here the first thing we need to do is, can we get the same thing as our denominator? Yes, we can. Also, that means we can multiply this by 3 over 3, 2 over 2, okay? So let's start from there. and see where it goes, so we multiply this by raising this to the power of 2 over 2 multiplied 1 over 10 plus x to the power of 3 is 3 over 3, sorry, 3 over 3, yes, so this is 2 over 2. is equal to 84, okay, so we have this as x to the power of 3 multiplied by 1 over 3 multiplied by 10 plus x squared multiplied by 1 over 2 multiplied by 10 plus x to the power of 1 over 30 is equal to 84. No, you're going to see what I'm trying to do. All right, so this will give us x cubed multiplied by 1 over 30 plus x squared multiplied by 1 over Sorry, we have 15 here. Oh my god, when did I change it to 10? So we now have to raise 1 to the power of 1 over 30 as well, plus x to the power of 1 over 30 is equal to 84. Now recall that when you have a to the power of m multiplied by n, that is equal to a to the the power of m raised to the power of n and also equals a to the power of n raised to the power of M, so it means that any of this power can be brought inside, and the other one can be brought in as well. outside, so that is exactly what we are going to do now, okay? Yeah, so we have x to the power of 1 over 30, let's bring that inside raised to the power of 3 since that is what we have in all of them so plus x to the power of 1 over 30 raised to the power of 2 plus x to the power of 1 over 30 is equal to 84 now if this is the first time you've ever seen this lovely face Click the subscription button and turn on your notification bell so that you don't miss our Amazing videos—we upload videos every single day, so you can be confident you'll see something amazing. New here, give us a thumbs up if you're actually enjoying this video and comment let's Actually, I know how you feel about what we're doing. Thank you very much. Now you can see that. We can actually simplify this equation if we let x to the power of 1 over 30. be equal to something, okay, let's call it y. Now, if that is the case, that means that this is y to the power of 3, so we have y to the power of 3 plus y to the power of 2 plus y to the power 2 plus y, right? Plus y is equal to 84. Are you seeing something? Yes. Now look at this. If this we Is y to the power of 3 times y to the power of 2 the same as y to the power of 1? Now, is there a number that when we raise it, we can raise that number to the power of 3, raise the number to the power of The power of 2 and raising the same number to the power of 1, when we add the three together, we get 84. So if you try, you notice that the four can work, that's right. So we have y to the power of 3 plus y. to the power of 2 plus y equals now 4 to the power of 3 will give us 64, right? 4 to the power of 2 will give us 16, and 4 to the power of 1 is 4; if we add everything here together, we are going to get 84, okay, so we have y cubed plus y squared plus y is equal to and 4 to the power of 3 plus 4 to the power of 2 plus 4. All right, now let's bring everything to the other side so that.unpack this further, all right, now we have y to the power of 3, let's bring this in; it becomes minus 4 to the power of 3, then we have plus y to the power of 2, if we bring this in, it becomes minus 4 to the power of 2, then we have plus y if you bring ...and as a result, we get negative 4 events. Now everything belongs to zero. Now, we need to group them. So we need to group them. We group them and we group them, okay? Now remember that y to the power of 3. minus okay this is an identity okay identity that a to the power of 3 minus b cubed is equal to a minus b multiplied by a squared plus a b plus b. Squared, okay, that's it for a to the power of 3 minus b to the power of 3 now y sorry a squared minus b squared is equal to a minus b multiplied by a plus b Now we are going to express this in terms of this and this in terms of this. Okay, so let's do that. Now we continue. Let me clean this well so that the board will be neater, okay? All right, now we y to the power of 3 minus 4 to the power of 3, so it's going to be y minus 4 times y minus 4 multiplied by y squared plus 2 a b, which is 2 multiplied by 4, sorry, times a b, which is 4 times y which is 4y, then plus b squared, that is 4 squared, which is 16. Okay, that is it for this one. then plus now y squared minus 4 squared is going to be y minus 4 multiplied by y plus 4 Okay, now let's clean the remaining parts of the board. Okay, all right. All right, now we have plus y plus 4, right? Sorry, y minus 4 is equal to 84. I'm sorry, I cleaned that. Okay, but remember that the remaining one is y minus 4, all right? Now we have y minus 4 here. y minus 4 here, y minus 4, let's factor that out; we have y minus 4, open bracket, when this is divided by this, this will remain, that is y squared plus 4y plus 16, and when this is divided by this y plus 4 remainder plus y plus 4, and when this is divided by this plus 1, we remain okay is equal to 84 now let's simplify further y minus 4 open bracket we have y squared then 4y plus y we give us 5y right, so we have plus 5y. Now 16 plus 4 is 20, and 20 plus 1 is 21, so we have plus 21 is equal to 84, now let us look at why I am still saying it is equal to 84, it is equal to 0, right? Yes, because Remember that we brought the 84 into this place after factoring it into 64 plus the other. numbers plus 16 plus 4, so this is equal to 0, equal to 0, equal to 0. I'm sorry about that, please. so you have 0 here, you have 0 here, and we have 0 here. All right. Now when you have a multiplied For a times b to be equal to 0, it simply means that either a is equal to 0 or b is equal to 0. Do you understand? Okay, now let's continue. Okay, all right. So from here, then we can say y minus 4 is equal to 0. or y squared plus 5y plus 21 equals 0. All right, now the next thing we need to do is to solve. the equations from the left-hand side; now remember that we are not looking for the value of y, we are Looking for the value of x, so we're going to obtain our x from y, so stay tuned and see how we got that. and also how we proved that the answer we obtained is correct, okay, so we have here that y is equal to minus 4 we move to become plus 4; now for this one, let us first of all check if we are won't have any real value from here, so let's talk about our determinant now, the determinant is b squared minus 4ac, so the determinant is equal to b, which is the coefficient of y, and that is 5. so we have 5 squared minus 4 multiplied by our a, which is the coefficient of y squared, which is 1, our c is our constant, which is 21, so we have that the determinant is 25 minus 4 times 21, which is 84. Okay, so when you solve this, you're going to have 4 minus 8 minus 5, which isn't possible; 14 minus 5. We'll give us 9 now, 7 minus 2 will give us 5, so we have minus 59, so you can see that this is less. greater than zero, which simply means that we will not obtain any real root from here. Okay, so let's move on. on this side, since we cannot obtain any real root from here, it simply means that the only value of y what we can use is y equals 4, but remember, recall that our x, that our y equals x. to the power of 1 over 30, yes, remember we got 1 over 30 from everything; I will say let x to the the power of 1 over 30 is equal to y, so it simply means that we replace y with 4, so it means that Now, in order to get x, we need to raise both sides to the power of about 30, okay. so we have 4 to the power of 30 is equal to x to the power of 1 over 30 raised to the power of 30 Okay, now remember that a to the power of m raised to the power of n is equal to a to the of m times n, so it means that this can be written as 4 to the power of 30 is equal to x to the power 1 over 30 multiplied by 30, okay, so we have 4 to the power of 30 is equal to this; we define this. to give us 1, so now we have the value of x to be 4 to the power of 30. Now let us prove that the Correct. So our x is 4 to the power of 30. Now let us prove that the answer we obtained is 4 raised to the power of 30, raised to the power of 1 over 10, plus 4 raised to the power of 30 4 raised to the power of 1 over 15 plus 4 raised to the power of 30 raised to the power of 1 over 30 Is this equal to 84? So you know that this is this multiplied by this, right? So we divide this to get 3, so we have 4 to the power of 3 plus now this we divide this to give us 2, so we have 4 to the power of 2; now this will divide itself to give us 1, and we have Just 4 is equal to 80 now; 4 to the power of 3 is 64, we have 64; 4 to the power of 2 is 16. And we have 4. Okay, now is this really equal to 84? Okay, now 16 plus 4 is 20, and 20 plus 64 is 84. So you can see that the answer is correct. Thanks so much for watching. See you in my next video. Goodbye