A Stanford University interview algebra math question that many failed

Hi everyone, let's find the square root of this without using a calculator. So first, we are going to simplify this further. So we have the square root of... Now we have 4, 4, 4, 4. And we have 1, 2, 3, 4, 5, 6. Let's replace that with six zeros. Then we add all of this. We have not changed anything. Our 2 is 5, a number. So we can further simplify this as... Square root of 1, 2, 3, 4. Then we have 6. 1, 2, 3, 4, 5, 6. Now this ends as 2, 2, 2, 2. 2, 0, plus 5. Now we have the square root of... This ends in 4, 4, 4, 4. Times 10 to the power of 1, 2, 3, 4, 5, 6. Then plus... This ends in 2, 2, 2, 2. 10 to the power of 1 Plus 5. Now we can express 4, 4, 4, 4, 4. As 4 multiplied by 1, 1, 1, 1. So we have 4 multiplied by 1, 2, 3, 4. Then multiplied by 10 to the power of 6. Besides, this ends as 2 multiplied by 1, 1, 1, 1. Into five places. 10 to the power of 1 plus 5. Remember that we are not asked to use the calculator. So we are trying to simplify this as much as possible. Now remember that 1, 1, 1, 1. That is 1,111. The same as 9,999 divided by 9. That's what we're going to do now. Remember that this is the square root. So all we are trying to do is bring in a perfect square. And notice that 9 is a perfect square. So we have the square root of... Now this is 4. So it's going to be 4 multiplied by... So 1, 1, 1, 1 now is... 9, 9, 9, 9. Divided by 9. Then multiplied by 10 to the power of 6. Then this is 2 brackets. 9, 9, 9, 9, 9. This one has five 9s. Divided by 9. Then times 10 plus 5. So we have the square root of... 4 brackets Now 9,999 can be written as... 9,999 So we have 10,000 minus 1. Divided by 9. We haven't changed anything. Then multiplied by 10 to the power of 6 plus 2 parentheses. Now this is the same thing as 100,000 minus 1. Divided by 9. Then multiplied by 10 plus 5. Alright, if this is the first time you've seen us, Click the subscription button and turn on your notification bell so you don't miss our amazing videos. We upload our videos every day. Two to three times every day. Tune in during your free time to see our new uploads. And give us a thumbs up if you're actually enjoying this video. Thank you very much. So back to what we are doing. Sorry, the root sign is down. It covers everything. Alright, so now we have the square root of... 4 brackets Now 10,000 is the same thing as 10 to the power of 4. 10 minus 1. Divided by 9. Multiplied by 10 to the power of 6. Plus 2 brackets. 100,000 Minus 1. Divided by 9. Times 10 plus 5. Now let's continue from here. This is actually interesting. Very, very interesting. Trust me. The logic in it had everything for me. Okay, so we have the square root of 4 multiplied by Now let's open the brackets. 10 to the power of 4 multiplied by 10 to the power of 6. 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. 10 to the power of 4 multiplied by 10 to the power of 6 10 to the power of 4 plus 6, which will give us 10 to the power of 10. 10 to the power of 4 multiplied by 10 to the power of 6 will give us 10 to the power of 10. 10 to the power of 10. So I'll write 10 to the power of 10 minus. Now 1 times 10 to the power of 6 is still 10 to the power of 6. Alright, now, plus two brackets, ten to the power of ten, remember that this is ten to the power of one, right? So ten to the power of five times ten to the power of one will give us ten to the power of six. ten to the power of six, minus one times ten to the power of one is still ten, okay? Then plus five, everything divided by nine. Now we have the square root of four multiplied by ten to the power of ten, I'm opening the brackets now, minus four multiplied by ten to six to the power of six, then plus two times this is two multiplied by ten to the power of six, then two Times minus ten is minus twenty, then plus five, okay? Now divide everything by nine. Are you Enjoying this? So we have the square root of, remember that we have here four multiplied by ten to the power of ten, okay? Now minus four times ten to the power of six plus two times ten to the power of six, we give us two minus four is minus two, right? Times ten to the power of six, then minus twenty plus five Everything, oh, oh, oh, oh, oh, that was too fast, no, this is not right, this is not right, sorry. We are still going to remember our, this is what we expanded to get, so it's from here to here, we give us over nine, from here to here is over nine, but definitely this is not over nine, this five, remember that it's on its own, right? I'm sorry about that. So this is what we have here—we retain that, we maintain that here. If we want to make this greater than nine, sorry, greater than nine, if we want to make this five also divisible by nine, we need to Multiply this five by nine, right? Yeah, so we have, it's going to be four times ten to the power of ten minus two times ten to the power of six, that is this minus this, right? Or before that, before that, this is what we are going to do. Let us make everything greater than nine so that we can then begin simplifying the numerator, okay? So we the square roots of four times ten to the power of ten minus four times ten to the power of six plus two multiplied by ten to the power of six minus twenty, then plus, then this five, I'll multiply it by nine to Give me forty-five. Yes, now at this point, I can make everything divisible by nine. Alright, I just hope I did not confuse some of you, okay? So that's it. Have you given us a thumbs up yet? If not, please do that for us. Okay, now It's time to factorize and simplify the numerator. So here we have four times ten to the power of ten, four times ten to the ten to the power of minus four minus four times ten to the power of six plus two times ten to the power of six gives us minus two times ten to the power of six. So we have minus two times ten to the power of six, then for five minus twenty, we get plus 25, everything divided by 9, and we still have a square root sign, okay, so this sentence The square root of 4 can be replaced with 2 squared, right? So, 2 to the power of 2. minus 10 to the power of now 10 to the 10 can be written as 5 times 2 so 5 multiplied by 2 Minus now here we have 2 times 10 to the power of 6 plus but 6 can be written as 5 plus 1, okay then we have plus 25 can be written as 5 squared, everything divided by 3 squared, or for now let's Just write 9, okay? I'm driving, so I'm trying to bring out um, a minus b all squared, you know what? I mean, yeah, so this is the square root of, now when you have a squared multiplied by b squared This is equal to a multiplied by b squared. Okay, so let's apply that here, so this Is 2 times 10 because this is 2 squared multiplied by 10 to the power of 5 squared, right? Yeah, all. -2 times this sentence 10 to the power of 5 multiplied by 10 to the power of 1 using the law of indices, yes, then plus 5 squared, everything divided by 9, we're getting there so this is the square root of now we have 2 multiplied by 10 to the power of 5 all squared Yeah, remember what I explained, all right? Then minus this sentence: 2 multiplied by 10 to the five to the power of 5 multiplied by 10, then plus 5 squared, all divided by 9. Now let's continue from Here we are, almost there. We're almost done. Yeah, we're almost done. Okay, so let's continue. so this time we have the square root of 2 multiplied by 10 raised to the power of 5 sorry 2 to the power 2 times 10 to the power of 5 squared, all right then minus this 10 can written as 2 times 5, so let me write it 2 brackets 2 times 10 to the power of 5 times 5 Right, yeah, 2 times 5 is still 10, then plus 5 squared, everything divided by 9. Yeah, perfect. This is exactly what I've been looking for, yes, because remember that a minus b all squared is equal to a squared minus 2ab plus b squared. Now in this case, our a is what: 2 times 10 to the power of 5. and our b is equal to 5; you can figure that out if you substitute this into this, you're You're going to get this right, okay? You're going to get this, okay? Now we have a minus b all squared. so that means we can replace all of this with, let me continue from here, yeah that means that we can replace the entire expression with the square root of a squared minus b squared Is 2 times 10 to the power of 5 minus 5, all squared, divided by 9? Okay, I say something, yeah. I believe so, so this is equal to the square root of 2 times 10 to the power of 5 minus 5, all squared. divided by now 9 squared is the same thing as 3 squared, 9 squared is the same thing as 3 squared, so we have 3 squared, okay, so this is equal to the square root of. Remember that when you have a squared divided by b squared is actually equal to a divided by b all squared, okay, so that means that we can write this as 2 times 10 to the power of 5 minus 5 divided by 3, all squared. Now remember that square we always cancel your square root sign, so that means that what we now have here is the square root, the square root, oh no, square root again, so we have 2 times 10 to the power of 5 which is the same thing as 2 times 10 to the power of 5, 1 2 3 4 5, that is 200,000 minus 5 divided by So we have this to be equal to 200,000 minus 5, which should be 199,995, right? 199,995 divided by 3. So if we do that, 3 goes into 19 five times with a remainder of 4. Okay, no—6, 6, 6, 6 with a remainder of 1. divide another 19 6 6 remainder 1 another 19 6 6 remainder 1 another 19 6 6 remainder 1 Now 15 will give us 5, and this is the answer. Yes, this is how to get the square root of a large number. Number like this using without using your calculator. Thank you so much for watching. I'll see you in my next video