A Nice square root math problem| solve for x

Hello, let's solve this using two methods. Welcome back to my channel. Today we have an amazing square root problem to solve and we are asked to solve for x if square root of x square root of x square root of x square root of x equals 5. Now first method let us square both sides. That is the first method. The second method is an amazing one and I would like you to watch the end and see that. So we have square root of x square root of square root of x then square root of square root of square root of x and then square root of square of x okay is equal to 5. Now if we square both sides this is what we will have right. Okay now remember the square root of x squared is equal to what x. Okay so that means that this is going to remove this right. So we have x then square root of x which is this then square root of square root of x and then square root of square root of x. This is what we'll have remaining is equal to 5 squared is 25. All right we have to square again. Okay so let us square again. So if we square again we will have or to avoid giving myself too many problems here or a large number to solve here let's maintain this as 5 squared. So what we have here is 5 squared all squared. Okay I'm going to do everything at once. All right now remember that a multiplied by b all squared is equal to a squared multiplied by b squared. Okay so that means that this is same thing as x squared multiplied by the square root of x square root of x square root of x squared. Okay is equal to 5. Now remember that a to the power of m raised to the power of n is equal to a to the power of m times n. So that means this will give us a 5 to the power of 2 multiplied by 2. All right so we have x squared multiplied by now this we remove this right so we have x then root x root x is equal to 5 to the power of 4. Then remember that a to the power of m multiplied by a to the power of n is equal to a to the power of m plus n. Okay so we are going to apply this here we're going to apply it here and if we do that we have this same thing as x to the power of one right. So we are going to have x times x we are going to pick one x and we add the powers 2 plus 1 which is 3. Okay then we have root x root x remaining is equal to 5 to the power of 4. All right now we are going to square again you know that our intention is to get x and the only way to remove your square root anytime you see it in an equation is to square it. Okay if this is the first time you're seeing this lovely channel please hit the subscription button and turn on the notification bell so that you don't miss our videos at all. All right give us a thumbs up if you love what we are doing. Thank you very much to all my subscribers, all my returning viewers. I appreciate you. Okay shout out to us to our 100,000 amazing subscribers. I love you guys so much. Thank you for the support for those of you that have been there all this while. May God keep blessing you for me. Thank you all I appreciate. All right now let's all square both sides again. So if we do that we have this squared and we have this squared. Okay so this will give us x to the power of 3 multiplied by 2 because it's going to be 3 squared. Okay yeah then this part squared multiplied by square root of x then square root of x squared is equal to 5 to the power of 4 multiplied by 2. So here we have x to the power of 3 multiplied by 2 which is 6 then multiplied by this we remove this and we have x root x okay is equal to 5 to the power of 8. Okay we are almost there. We are almost there. All right so we have this is going to be we are going to solve this using the same method just like we did here. So this will give us x to the power of 6 plus 1 is 7. Okay then we have root x is equal to 5 to the power of 8. Then we are going to square again so that we remove the last square root. Okay so if we square again we will have x to the power of 7 squared then root x squared is equal to 5 to the power of 8 multiplied by 2. So we have x to the power of 7 times 2 is 14 then this we remove this and we are left with just x is equal to 5 x times 2 is 16. So this same thing as x to the power of 1 so we'll have x to the power of 14 plus 1 right is equal to 5 to the power of 16. So we have x to the power of 15 is equal to 5 to the power of 16. All right now we need to get the value of x and the only way to do that is to remove this power of 15 and the only way to remove the power of 15 is to multiply it by to raise this side to the power of 1 over 15. So because of that we are going to raise both sides of this equation to the power of 1 over 15. So if we do that we have x to the power of 15 raised to the power of 1 over 15 is equal to 5 to the power of 16 raised to the power of 1 over 15. So 15 we remove 15 and we have x is equal to 5 to the power of 16 multiplied by 1 over 15. So we have x is equal to 5 to the power of 16 over 15 and that is our x. All right now let us solve this using the second method. Like I said the second method is the most interesting method. So let's go ahead and solve this using the second method. Now for the second method, remember that this is same thing as square root of x. Before that let me write this that square root of a multiplied by b is equal to square root of a multiplied by square root of b. So that means that if I write this as this square root of x multiplied by square root of square root of x multiplied by square root of square root of square root of x then multiplied by 4 square root of x right? Square root of square root of square root of square root of x. So we have this is 1, 2, 3, 4. Okay is equal to 5. Now this is the, if we expand this, this is exactly what we have. But remember that square root of x is the same thing as So that means that we can write this as x to the power of half multiplied by x to the power of half raised to the power of half multiplied by x to the power of half raised the power of half, raised to the power of the third half. Then multiplied by x to the power of half, raised to the power of the second root, raised to the power of half the third one, then raised to the power of half the fourth one, is equal to 5. Now remember that a to the power of m, raised to the power of n, is equal to a to the power of m times n. Therefore, this is the same thing as x to the power of half right, multiplied by x to the power of half times half right, which will give us 1 over 4, multiplied by x to the power of half times half times half, 1 times 1 times 1 is 1, over 2 times 2 is 4, 4 times 2 is 8, multiplied by x to the power of 1 over 2 times 2 is 4, 4 times 2 is 8, 8 times 2 is 16, is equal to 5. Alright, I hope you are enjoying this series, because I am enjoying it. Now, remember that a to the power of m, multiplied by a to the power of n, is equal to a to the power of m plus n. So that means you are going to take one root, which is x, and one base, which is x, and we are going to add the powers. So we have 1 over 2, plus 1 over 4, plus 1 over 8, plus 1 over 16, is equal to 5. So we have x to the power of, now the LCM of 2, 4, 8, 16, is 16. Now when 2 divides 16, we have 8, 8 times 1 is 8. When 4 divides 16, we have 4, 4 times 1 is 4. When 8 divides 16, we have 2, 2 times 1 is 2. And when 16 divides itself, we have 1, 1 times 1 is 1, is equal to 5. So we have x to the power of, 8 plus 4 is 12, 12 plus 2 is 14, 14 plus 1 is 15, so we have 15 over 16, is equal to 5. Now to get the value of x, we are going to raise each side of this equation to the power of 16 over 15, the inverse of this. So we have x to the power of 15 over 16, multiplied by 16 over 15. So we are raising it, sorry, is equal to 5 to the power of 16 over 15. Now, remember that, a to the power of m, raised to the power of n, is a to the power of n times n. So this same thing as x to the power of 15 over 16 multiplied by 16 over 15, is equal to 5 to the power of 16 over 15. So this, we remove this, and this, we remove this, and we have that our x is 5 to the power of 16 over 15. Thank you so much for watching, and see you in my next video. Bye.