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

Added: 2026-05-24

[00:00:00]entrance exam question. Let's simplify this. Welcome back to my channel.

[00:00:04]First things first, let us simplify the numerator. The numerator can be written as root six multiplied by three.

[00:00:16]Then this can be written as root six multiplied by two.

[00:00:20]Then we have root six.

[00:00:23]Everything raised to the power of 10.

[00:00:26]Then we can split this because [clears throat] square root of A multiplied by B is same thing as square root of A multiplied by square root of B, okay? So we have square root of six multiplied by square root of three divided by square root of six multiplied by square root of two minus square root of six.

[00:00:46]Everything to the power of 10.

[00:00:50]So going forward, we can factorize this from the denominator. So we have root six multiplied by root three divided by root six bracket. This will give us root two take away one.

[00:01:08]Now everything to the power of 10, okay?

[00:01:11]So this is equal to you can see that this and this can go and we are left with root three divided by root two take away one to the power of 10.

[00:01:23]Now let's split this.

[00:01:26]We can write this as root three raised to the power of 10 over root two take away one raised to the power of 10.

[00:01:35]This can be further simplified as square root of a number is that number to the power of half. So we have three to the power of half raised to the power of 10 over root two minus one raised to the power of 10. So this will remove this, we have five. That is because when two powers are on a particular number, you do what? You multiply them together. Yes. So, we have 1/2 * 10 which will give us five.

[00:02:05]So, moving forward now, we have 3 to the power of five divided by root two minus one raised to the power of 10. Now, 3 to the power of five is 243, right?

[00:02:19]243 divided by root two minus one to the power of 10.

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[00:02:50]So, I'm going to copy the last part which says that this is equal to 243 over root two minus one to the power of 10. Now, um looking at this, the numerator is already simplified. So, let's dive into the denominator and simplify it further.

[00:03:17]So, we can say let root two minus one be equal to X.

[00:03:22]This implies that This implies that root two minus one to the power of 10 cannot be expressed as X to the power of 10, right?

[00:03:34]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, okay? 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.

[00:04:01]Then, we need to remove this root sign.

[00:04:03]So, let's square both sides of this equation.

[00:04:08]So, we have X plus 1 squared is equal to root 2 squared.

[00:04:15]Now, remember that A plus B squared is [snorts] equal to A squared plus 2 AB plus B squared.

[00:04:25]All right?

[00:04:27]So, we can rewrite this as X squared plus 2 X plus 1. That is for this. Is equal to This will remove this, we have two.

[00:04:41]All right? Um someone corrected me about this in the comment. He said I should stop canceling it out and explain why.

[00:04:48]Now, this is why this and the root sign and the square cancels out. This is same thing as two to the power of half, right? Then, raised to the power of two.

[00:04:58]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 same thing as two to the power of half multiplied by two over one. So, you can see that this and this will go and you are left with just two.

[00:05:13]I always skip this because I feel it's wasting our time more, all right?

[00:05:22]Now, moving forward, we 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 2 X minus 1.

[00:05:38]So, I have X squared is 2 Oh, I don't think you can see this side.

[00:05:45]So, I now have X squared is equal to 2 - 1 is 1. So, 1 - 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 uh powers we are going to be getting.

[00:06:10]All right? So, we continue.

[00:06:18]So, remember that we are looking at solving or simplifying x to the power of 10.

[00:06:26]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 - 2x all squared.

[00:07:02]So, we have x to the power of 4 is equal to This will give us 1 squared 2 multiplied by -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 - 4x + 4x squared.

[00:07:30]Okay. Now, we have x squared here. We are going to substitute what we got as our x squared, which is uh Let me not write it again. So, we have 1 - 2x is to squared. Very important equation.

[00:07:47]So, I'll leave it here and because I'll always refer to it.

[00:07:52]So, we have x ^ 4 is 1 - 4 x + 4 brackets. Our x squared is 1 - 2 x.

[00:08:01]x ^ 4 is 1 1 - 4 x + 4 * 1 is 4. 4 * this is - 8 x.

[00:08:10]So, x ^ 4 is 1 + 4 is 5 - 4 x - 8 x is - 12 x.

[00:08:19]Now, we are going to square this again.

[00:08:22]Remember what I explained, okay?

[00:08:25]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 >> [snorts] >> is equal to 5 squared - 2 * 5 * 12 x.

[00:08:56]Okay? Then, + 12 x squared.

[00:09:02]So, we have x ^ 8 is 25 120 x + 144 x squared. It has gotten to the point where I need to be careful so that I don't make mistakes >> [laughter] >> with the numbers. But, if I see if I do, just pardon me, all right?

[00:09:23]So, I have x ^ 8 is 25 - 120 x + 144.

[00:09:31]Now, our x squared is 1 - 2 x. So, we are substituting that.

[00:09:38]We have x to the power of 8 is 25 - 120x + 144 * 1 and minus this will be 288x.

[00:09:49]>> [snorts] >> Time to join together, we have x to the power of 8 is then this will give us 169 minus 8 [snorts] + 0 is 8 8 + 8 10 carry 1 2 + 1 is 3 + 1 4.

[00:10:09]Time to be very careful so that I don't make mistake.

[00:10:12]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. Yeah. So, we need to multiply both sides.

[00:10:27]This time I want to multiply. Multiply both sides by x squared.

[00:10:33]Okay? So, I'm multiplying this time and 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.

[00:10:48]It should give me the same thing. Right?

[00:10:51]Right. So, >> [laughter] >> if you watched to this stage, please leave me a love emoji on the comment.

[00:11:02]That will show me that you actually watched to this end and I'm going to give you a special thank you or comment for that. All right?

[00:11:11]So, this is what we have. We are multiplying both sides by x squared.

[00:11:16]So, we have x to the power of 8 multiplied by x squared is equal to x squared multiplied by 169 minus 408x.

[00:11:30]Now, you can know you know that this multiplied by this will give us x to the power of 8 + 2, right? Which is 10.

[00:11:38]Is equal to now our x squared is 1 - 2x.

[00:11:42]So, we have 1 - 2x bracket 169 - 408x. So, x So, x to the power of 10 will give us 1 * this is 169.

[00:11:55]1 * this is -408x.

[00:11:59]I'm trying not to make mistake.

[00:12:01]So, pardon me if I become slower here.

[00:12:04]Now, 169 * 2 2 * 9 18 2 * 6 12 13 carry 1 2 * 1 2 + 1 3. So, we have 338.

[00:12:15]So, now we have minus 338x.

[00:12:20]Okay?

[00:12:22]Then, we have minus times minus is plus.

[00:12:26]Then, 408 * 2 is same thing as 800 + 16.

[00:12:32]So, it's going to be 816, right? So, 816x squared.

[00:12:40]816x squared. Good.

[00:12:44]Hope I've not made a mistake. So, we continue.

[00:12:49]We have x to the power of 10 is 16 9. Now, minus this minus that. Let's solve it. 408 338 16 carry 1 4 846.

[00:13:07]So, we have 846x.

[00:13:12]Plus 816x squared.

[00:13:15]So, we have x to the power of 10 is 169 - 846x + 816. Now, remember our x squared is 1 - 2x.

[00:13:26]1 - 2x.

[00:13:29]Now, x is 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 1632 x okay So now we have 169 plus 816 This is 5 This is 8 985 So we have x to the power of 10 is equal to 985 985 then minus this minus that let's add them together 846 1632 This is 87 14 carry 1 2478 So minus 2478 8x Now we have x there and we are not leaving it there >> [laughter] >> So we have x to the power of 10 is 985 minus 2478 bracket Remember that our x is what root 2 minus 1 So we have [snorts] this to be equal to 985 minus 2478 root 2 plus 247 8, right?

[00:15:36]Yeah.

[00:15:37]Okay, now we have to add this to this.

[00:15:41]If I make mistake anywhere, I'm not coming back to redo this video.

[00:15:47]Just use your calculator and correct it.

[00:15:49]Please.

[00:15:51]If I make any mistake anywhere, use your calculator and correct it in the comment section. Simple.

[00:15:59]Please.

[00:16:02]So, we continue. 8 + 5 is 13.

[00:16:07]We carry one.

[00:16:10]9 + 8 is 17. Carry one.

[00:16:13]9 + 5 is 14. Carry one. So, we have 3473.

[00:16:18]3473.

[00:16:20]So, this is equal to 3473 minus 2478 root 2. That is x to the power of 10.

[00:16:29]So, this simply means that um we can write 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 denominator. So, this can be written as 243 divided by 3473 minus 2478 root 2.

[00:17:09]>> [clears throat] >> If I don't make mistake anywhere. Thank you for watching and see you in my next video. Bye.