Only 2% correctly figured out this math question

Hello everyone. Happy to see you here. Welcome back to my channel HiMathematics. Today we have a very interesting and tricky question, square root of 16 minus square root of 12. And a lot of students might be thinking, maybe we can subtract 16 minus 12, as a result we have square root of 4. And to be honest, they go in the wrong direction. So how can we solve this question correctly and step by step? First of all, let's simplify this real quick. Square root of 16 can be easily written as 4. This is our table K, so let's start with that. So square root of 16 can be written as 4. So as a result, we have 4. This 12, how can we write, how can we express this 12? We have like three combinations. We have 2 times 6. This is great, I guess, but better option for us is 4 times 3. We can easily split it and as a result, we can find square root of 4. The third option for us is 1 times 12. This is not a good option. So as a result, let's choose this second option, 4 times 3. So this square root of 12, we can write as 4 times 3. So minus square root of 4 times 3. And as I said before, according to a basic square root property, I'm going to write this property right here on the bottom. When you have square root of A times B, when you have a product, it's not worked with addition and subtraction. It works only with division and multiplication. So when you have square root of A times B, we can easily split it like square root of A times square root of B. I really hope you get it. And right now, let's split this expression. So as a result, what do we get? Square root of, we have 4 minus square root of 4 times square root of 3. I really hope you understand it. So equal sign, equal sign right here. Okay, right now, square root of 4 equal to 2. This is all known table case, so let's start with that. So square root of 4 minus 2 square root of 3. And right now, what is the main idea? Because right now, we can easily see that we can't do simplification, we can't simplify it, we have square root of 3, we can't find it, we have 2, we have 4, so we can't work with it. But what is the main idea? The main idea is to create a square inside of a square root sign. So what I mean right now, when you have square root of, and inside square root, you have A, for example, square. Everyone knows that we can easily cancel this square root sign and these two. As a result, we have not A, but absolute value of A. This is a correct way how can we express it. So when you have, for example, square root of 3 square, everyone knows that we can easily cancel this square root sign and two, and as a result, we have 3. Okay, and right now, let's use the same logic. Let's try to create square inside of a square root sign. How can you do this? I'm going to show you a really interesting trick. So instead of this 4, let's write 3 plus 1. Okay, instead of 4, we're going to write 3 plus 1. So this 4 can be written as 3 plus 1. Let's do this. We have 3 plus 1 and minus 2 square root of 3. So I really hope you understand it. What are we going to do next? Right now, let's write this 1 on the third position. So on the first position, 3. So it changed nothing. We can easily change position right here. This is not multiplication and division. Yeah, so we have 3. The next thing we have minus 2 square root of 3. So this one, we're going to write on the second position. So minus 2 square root of 3 and plus 1 and plus 1. And a lot of students ask me why I do this. What are we going to do right now? What is the main purpose? What is the main idea of doing it? Right now, I'm going to show you two really important tricks. First of all, instead of this 3, let's write the next expression. So instead of this 3, I'm going to underline it right here. So instead of this 3, we're going to write square root of 3 square to the second power. This is equal to 3. We change nothing. What are we going to do with the second expression? We write a square root of 1. So we multiply this expression by square root of 1. It's also changed nothing. So we have minus the same thing, 2 square root of 3. And right now, I'm going to multiply it by square root of 1. Don't ask me why. In the few seconds, you're going to understand it. So square root of 3, we're going to multiply it by square root of 1. It changed nothing because square root of 1 equal to 1. And 1 times this expression equal to 2 square root of 3. And instead of this one, we're going to write square root of 1 to the second power. Square root of 1 to the second power. So right now, I'm going to underline what I did before. So first of all, instead of this 3, I write square root of 3 to the second power. It changed nothing. Second expression, I multiply it by square root of 1. It's also changed nothing. And the third expression, I write with this structure. So square root to the second power. It's also equal to 1. But right now, let's look closely at what we have right here. I'm going to write a really important algebra formula. Let's do this right now. I'm going to write it on the left side. So we have a minus b square. OK. I'm going to write this formula. I'm a minus b to the second power. This is equal to a square, a square minus 2ab and plus b square. And when I write this formula, a lot of students get it. A lot of students understand why I split it, why I expand this formula. Because in the same way, a lot of students might be thinking, Mr. you expand this expression. It's not the expression that we had in the beginning. So very easy for us. 4, 4, 3, square root. So it's not that hard. But right now, you expand it in a difficult way. Square root, squares. And right now, here's my answer to students. First of all, if you look closely at this formula, at this expression, we can easily underline that expression that we have in our formula. Because for example, square root of 3, this is our a. We have a square. For example, we have 2. This is really great. If this is a square, then square root of 3 is our a. When b is square root of 1, we can easily use this as square root of 1. Square root of 1 to the second power. This is our b square. So I really hope you understand it, that after this expansion of our expression, we get that our formula is also correct to express our expression. Because this is our formula, a minus b square. a equal to square root of 3. So as a result, right here, we can write square root of square root of 3. b equal to square root of 1. But we can easily write it as 1. So we have square root of 3 minus 1 to the second power. And this is the main purpose why we do this. Because right now, we can easily cancel our square root sign. So from here, we can easily cancel it. And from here, we can easily cancel it. As a result, this is a very tricky moment. Because a lot of students forget about it. As I said before, this is equal to absolute value a. OK. So as a result, we need to write this is equal to absolute value square root of 3 minus square root of 3 minus 1. And right now, how can we work with it? Because we don't need its absolute value. The main idea is the next one. We need to find out what is this expression. This is if this positive expression or negative expression. OK. Square root of 3 is greater than 1, obviously. OK. So square root of 3 is greater than 1. So it means that this expression is positive. So we can easily get rid of these absolute value brackets. So as a result, we have square root of 3 minus 1. And this is our answer, square root of 3 minus 1. We can't simplify this more. Maybe if you're interested in approximately answer, because a lot of students, if you're talking about approximately answer, a lot of students know that square root of 3 is approximately equal to 1.732, if you're talking about that. And of course, if you thought from here, so if you're talking about approximately answer, this expression, square root of 3 minus 1, is approximately equal to 1.732 minus 1, which is equal to 0.7732. So as you can see, after this solution, we can easily find approximately answer. So it's not needed. For example, if you simplify it on your exam, it's not needed because sometimes you have like square root of 5. So you can't know the answer. Maybe you can easily get it when you know approximately answer. So in our case, square root of 3 and square root of 2 is the easiest one, because for example, square root of 17, you don't know about it. So in our case, you can easily write our approximately answer. So let's write our final answer to our question. And after that, I'm going to give you a few really important hints, a few really important prompts to this question. So square root of 16, square root of 16 minus square root of 12 equal to square root of 3 minus 1. And if it's possible, you can write it that this is equal to 0.7732. And right now, I'm going to give you a few really important moments, because in this case, a lot of students might be thinking that we need to cancel all the time, and we don't need to write these absolute value brackets. But just imagine, for example, I'm going to write it right here on the bottom. For example, imagine you have square root of, and you have, for example, square root of 5 minus square root of 10. Let's write something like that to the second power. And a lot of students might be thinking, OK, we need to cancel this square root sign, these two. And of course, you can easily do this. And as a result, they say, OK, square root of 5 minus square root of 10, because we cancel it. We can easily solve it like that, because teachers said that when we have square root of, for example, 3 square, we can easily cancel it. And this is absolutely, absolutely correct. But when you have expression, I strongly recommend you to put absolute value brackets. OK, when you put absolute value brackets, the next step you need to do, you need to find out the value of this expression, like approximately, like the sign of this expression, positive or negative. In our case, square root of 5 is less than square root of 10. OK, so it means that this expression is a negative. So in our case, we need to put minus, we need to put parentheses, square root of 5 minus square root of 10. OK, and as a result, we can easily write it as square root of 10 minus square root of 5. And this is a correct approach to do this, with this principle, with this algorithm. Of course, if you work with the constant, in 99% of cases, you can find the square root of a positive integer, so in 99% of cases. But when you have expression, you do know the value of this expression, positive or negative. So put absolute value brackets, compare this expression, like find out what is the sign of this expression, and then after two more steps, you can get a correct answer, because this is a wrong answer without absolute value brackets. So I really hope you understand my explanation. I really hope you learned something new. Once more, I want to show you this solution to this question. It's not that hard. I really like this solution, because this is a great algebra question with basic algebra steps. We just split it, remember two properties. We just change it all a little bit. And the trickiest moment is right here, to see that we can expand it, we can write it according to this identity. This is like a trickiest moment right here. So if you get this moment, you can easily be proud of yourself that you can solve this type of question, because there are a lot of these combinations, you know, and when you know this principle, when you know this algorithm, you can easily be proud of yourself, that you know how can you solve it. I really hope I helped you with this video. I really hope you understand my explanation. If you do, you can easily write your question, write your assumption down into the comment section, and let's discuss a little bit about this type of question. And thank you for watching. Wish you all the best in life. Take care of yourself. Have a great day. See you in the next videos.