This Radical Math Question Had Me Stumped… For a Sec, My Brain Stopped Working!

Let's simplify this. Welcome back to my channel. First things first, let us simplify the numerator. The numerator can be written as root 6 multiplied by 3. Then this can be written as root 6 multiplied by 2. Then we have root 6. Everything raised to the power of 10. Then we can split this because square root of a multiplied by b is same as square root of a multiplied by square root of b. So we have square root of 6 multiplied by square root of 3 divided by square root of 6 multiplied by square root of 2 minus square root of 6. Everything to the power of 10. Going forward, we can factorize this from the denominator. So we have root 6 multiplied by root 3 divided by root 6 bracket. This will give us root 2 take away 1. Now everything to the power of 10. So this is equal to, you can see that this and this can go and we are left with root 3 divided by root 2 take away 1 to the power of 10. Now let's split this. We can write this as root 3 raised to the power of 10 over root 2 take away 1 raised to the power of 10. This can be further simplified as square root of a number is that number to the power of half. So we have 3 to the power of half raised to the power of 10 over root 2 minus 1 raised to the power of 10. So this, we remove this, we have 5. That is because when two powers are on a particular number, you multiply them together. So we have half multiplied by 10 which will give us 5. So moving forward now we have 3 to the power of 5 divided by root 2 minus 1 raised to the power of 10. Now 3 to the power of 5 is 243, right? 243 divided by root 2 minus 1 to the power of 10. Now if this is the first time you have seen this channel, please click the subscription button, turn on your notification bell so that you don't miss our amazing videos. We upload our videos as often as possible to your convenience. There must be something you will see to watch by God's grace. Give us a thumbs up if you love what we are doing. Thank you very much. So I'm going to copy the last part which says that this is equal to 243 over root 2 minus 1 to the power of 10. Now looking at this, the numerator is already simplified. So let's dive into the denominator and simplify it further. So we can say let root 2 minus 1 be equal to x. This implies that root 2 minus 1 to the power of 10 can now be expressed as x to the power of 10. Now when that happens, what do we do next? We are going to concentrate on this equation and we are going to simplify this equation. So first of all we are going to transfer this to the other side. But first permit me to write this as x is equal to root 2 minus 1. Now let's transfer this to this side. We will have x plus 1 is equal to root 2. Then we need to remove this root sign. So let's square both sides of this equation. So we have x plus 1 squared is equal to root 2 squared. Now remember that a plus b squared is equal to a squared plus 2ab plus b squared. We can rewrite this as x squared plus 2x plus 1 is equal to 2. Someone corrected me about this in the comment. He said I should stop cancelling it out and explain why. Now this is why the root sign and the square cancel out. This is the same thing as 2 to the power of half. Then raised to the power of 2. And when you have a to the power of m raised to the power of n, this is equal to a to the power of m times n. So that means this is the same thing as 2 to the power of half multiplied by 2 over 1. So you can see that this and this will go and you are left with just 2. I always skip this because I feel it's wasting our time more. Now moving forward, I need this x squared. You are going to see why. So I'm going to transfer every other thing to the other side. So I'll have x squared is 2 minus 2x minus 1. So I have x squared is 2 I don't think you can see this side. So I now have x squared is equal to 2 minus 1 is 1. So 1 minus 2x. Now this is the most important equation in this particular question because this is the only equation that will always help us to simplify the complex powers we are going to be getting. So we continue. So remember that we are looking at solving or simplifying x to the power of 10. And now we have x squared. To get to x to the power of 10 we need to square x squared to get x to the power of 4, square it again to get x to the power of 8, square it again to get x to the power of 10. I hope you understand what I'm saying. So we need to square x squared to get to this first of all. That is the main thing I'm trying to say now. So let us square both sides of this equation. So we have x to the power of 2 all squared is equal to 1 minus 2x all squared. So we have x to the power of 4 is equal to, this will give us 1 squared minus 2 multiplied by negative 2 x multiplied by 1 which is still the same then plus 2x all squared. So we have x to the power of 4 is equal to 1 minus 4x plus 4x squared. Now we have x squared here. We are going to substitute what we got as our x squared which is, let me not rewrite it again. So we have 1 minus 2x is equal to x squared. Very important equation. So I will leave it here because I will always refer to it. So we have x to the power of 4 is 1 minus 4x plus 4 brackets. Our x squared is 1 minus 2x. x to the power of 4 is 1 minus 4x plus 4 times 1 is 4, 4 times this is minus 8x. So x to the power of 4 is 1 plus 4 is 5 minus 4x minus 8x is minus 12x. Now we are going to square this again. Remember what I explained. So if we square this and square this. We are going to have x to the power of 8. So we have x to the power of 8 is equal to 5 squared minus 2 multiplied by 5 multiplied by 12x then plus 12x squared. So we have x to the power of 8 is 25 minus 120x plus 144x squared. This has gotten to the point where I need to be careful so that I don't make mistakes with the numbers. But if I do just pardon me. So I have x to the power of 8 is 25 minus 120x plus 144. Now our x squared is 1 minus 2x. So we are substituting that. We have x to the power of 8 is 25 minus 120x plus 144 times 1 and minus this will be 288x. Time to join together. We have x to the power of 8 is 169 minus 8 plus 0 is 8. 8 plus 8 is 10. Carry on. 2 plus 1 is 3 plus 1 is 4. Time to be very careful so that I don't make mistakes. So now we have x to the power of 8. We need x to the power of 10. Remember we need x to the power of 10. So we need to multiply both sides. This time I want to multiply both sides by x squared. So I'm multiplying this time. I'm not squaring both sides because I have large numbers here. Squaring it is going to give me hard time. So I'm multiplying both sides by x to the power of 2 which will give me the same 10. Right? Right. So if you watched to this stage please leave me a love emoji on the comment that will show me that you actually watched to this end and I'm going to give you a special thank you comment for that. Alright? So this is what we have. We are multiplying both sides by x squared. So we have x to the power of 8 multiplied by x squared is equal to x squared multiplied by 169 minus 408x. Now you know that this multiplied by this will give us x to the power of 8 plus 2. Right? Which is 10. Now our x squared is 1 minus 2x. So we have 1 minus 2x bracket 169 minus 408x. So x to the power of 10 will give us 1 times this is 169. 1 times this is minus 408x. I'm trying not to make mistakes. So pardon me if I become slower here. Now 169 times 2 2 times 9 is 18. 2 times 6 is 12. 13 carry 1. 2 times 12 plus 13. So we have 338. So now we have minus 338x. Okay? Then we have minus times minus is plus. Then 408 times 2 is the same as 800 plus 16. So it's going to be 816. Right? So 816x squared. Good. I hope I've not made a mistake. So we continue. We have x to the power of 10 is 169. Now minus this minus that. Let's solve it. 408338 16 carry 1 for 846. So we have 846x plus 816x squared. So we have x to the power of 10 is 169 minus 846x plus 816. Now remember our x squared is 1 minus 2x. 1 minus 2x. Now x to the power of 10 is 169 minus 846x plus this times this is 816. This times this is minus. Now we are going 816 times 2. This is 12. This is 2 plus 1 is 3. This is 16. So we have 1632. 1632x. Okay? So now we have 169 plus 816. This is 5. This is 8. This is 985. So we have x to the power of 10 is equal to 985. 985. Then minus this minus that. Let's add them together. 8461632. This is 8714 carry 1 2478. So minus 2478 8x. Now we have x there and we are not leaving it there. So we have x to the power of 10 is 985 minus 2478 bracket. Remember that our x is root 2 minus 1. So we have this to be equal to 985 minus 2478 root 2 plus 2478 Right? Yeah. Okay. Now we have to add this to this. If I made a mistake anywhere, I'm not coming back to redo this video. Just use your calculator and correct it. Please. If I made any mistake anywhere, use your calculator and correct it in the comment section. Simple. Please. So we continue 8 plus 5 is 13. We carry 1. 9 plus 8 is 17. Carry 1. 9 plus 5 is 14. Carry 1. So we have 3473. 3473. So this is equal to 3473 minus 2 478 root 2. That is x to the power of 10. So this simply means that we can rewrite this. Root 18 divided by root 12 minus root 6 raised to the power of 10 to be equal to this 243 divided by root 2 minus 1 to the power of 10 which is equal to. We already know that x to the power of 10 is this. The rest can be written as 243 divided by 3473 minus 2478 root 2. If I did not make a mistake anywhere. Thank you for watching and see you in my next video.