 Okay, great. So I'm getting a lot of answers. Question number three is decimal representation of a rational number cannot be. It can be terminating. It can be non terminating repeating. It can be non terminating also, but it can never be non terminating non repeating because if it is, I was just explaining this. If this is non terminating non repeating, then it will become an irrational number. It will never be represented in form of P by Q. So that's the answer. Second answer is sometimes the product of any two irrational number. Let's take root two into root three. So what does it give me? It gives me a root six. So this is irrational. But root three into root three gives me what three. So this is rational. So sometimes rational, sometimes irrational. The decimal expansion of number root two is a non terminating non recurring number because it's an irrational rational number, which of the following is irrational. So root four by nine will give me two by three. This will give me 12 by 12 by three, which is four. So this will give me two. This will give me nine. So this is my irrational number, seven. Now, as soon as you look at a bar in front of the number, sorry, above the number, it gives me an idea that the same number is getting repeated time and again. So here a point one four can be written as 14 by 100. So it is rational. This can be written as this can be converted into a rational number. I'll tell you in the further question. Similarly, this can be converted. So these are not my answer irrational number is this, because if you look at it, there is no proper repetition here. So this is not repeating itself. Now I was taking taking an example of root two and three. So root two and three, the rational number in between. So addition of two irrational number is is irrational. So this cannot be my answer. Now, root two into root three is again root six, which is irrational. So that this cannot be my answer. And root two is nearly 1.41 and root three is 1.73. So 1.8 is beyond that limit. So here answer is 1.5. So you can check the answer using that. So that's the answer. Now let's go to question number nine to 13. So for the first one, won't it be approximately? Yeah, obviously it would be approximately. But this is repeating, right? So this can be written in written in form of 1.9 bar. So there is no question of approximately you can convert it into proper values. Okay, so I'm getting few answers now. Okay, two more minutes for these questions, not more than that. Okay, I think I should solve these questions. First question is up to convert 1.999 where nine is repeating infinitely in P by Q form. So I can do it like this. So I assume X to be 1.999 where nine is repeating infinitely. So X is 1.9 bar. So what we do is generally the idea is that we get repeating portion once in front of the decimal. So here repeating portion is only nine and to get nine out of the decimal, we need 10. So I multiply it with 10. I get 10x is equal to 19.9 bar. And then I subtract this so 10x minus X is 9x and 19 minus one is 18.0 because nine bar and nine bar gets cancelled out. So this comes out to be 18 by nine and 18 by nine is equivalent to two. So the answer is two out here. So this particular part will give me an answer to now this is what this is two three plus root three. So if irrational parts of two irrational numbers are same, we add their rational portions. So if I have a root B plus C root B, it would be a plus C root B. So here I have two root three plus one root three, that would be two plus one root three, that is three root three. Then I have root 10 into root 15. So root a multiplied by root B is nothing but root AB. If root AB, I write it root 10 into root 15, which is equivalent to root 150. Now root 150, what I do is I prime factorize this number. So 150, if you divide it by two, you get 75. Then you divide it by three, you get 25 and 25 is five into five. So you can write two into three into five into five. So here whatever is inside the square root, whatever is repeating itself will come out once. So five is coming two times five will come out one. So it is five root six. So what do you get here? Five root six here. Now, so till now I am getting answers like this, it is C, C and B. Now let me move to other two questions. The other two questions are something like this. So here I have to rationalize it. Rationalization happens with changing or taking a conjugate of the sign in between. Conjugate means conversion of negative to positive or positive to negative. So I can do that. So I write one by root seven minus two multiplied by root seven plus two because here it is negative and root seven plus two. So in numerator, root seven plus two is there. In denominator, it is X minus Y multiplied by X plus Y format, which is X square minus Y square. So root seven square is seven minus two square is four. So this comes out to be root seven plus two by three, which is your answer A. So the next answer is A. And I saw a lot of you giving answer C because you didn't take two square. Now let me go to the same thing. Now look at here. There is only one place where in the denominator where there is a denominator, which is option B. So first I'll check option B whether option B is correct or not. Now I know that root nine is nothing but three and root eight is two root two. But if you look at the sign in between, the sign in between is negative. So it can be three minus two root two. It will not be three plus two root two. So what I do is I will multiply it with conjugate. So numerator is root nine plus root eight. Denominator is nine minus eight, which is equivalent to I can write numerator as three plus two root two divided by nine minus eight is one, which is option D. So it is CCBAD. That's the answer for these questions. Okay, now let me move to another type of questions. And let me give you some difficult questions now. So now let's move to some difficult questions. Okay, go to question nine, not question nine. Okay, question 11, can you solve? First part of question 11. Okay, I have got two answers till now. Someone is giving 59. Someone is someone is giving 11. Okay, should I solve this question? Enough time for this question, I think. Okay, let me solve this question. It is five plus two root three divided by seven plus four root three. So if it is five plus two root three divided by seven plus four root three, five plus three divided by seven plus four root three, the only way to remove denominator is to rationalize it. How do I rationalize it? So I multiply it with the negative in between because here it is positive. So I take opposite. So seven minus four root three. Now, let me first look at the denominator. The denominator is the denominator is seven square is 49 minus four square is 16 into 348. Five plus seven is 35 here. Five minus four root three is minus 20 root three. Two root three into seven is plus 14 root three. And two root three minus four root three is minus eight into three because root three into root three is three and two into four is eight. So what do I get? I get 35 minus 24, which is 11 minus six root three divided by one. So I'm not writing one. So if I compare it with a minus six root three, the value of K comes out to be equivalent to 11. So that's your answer. So now let me move to another question. And this question which I'm selecting is find out the value, find out question four here. And then I would also like you to solve question 12. No, first you solve question 11 fourth and then 12. Okay. Two, three more minutes. Okay, I'm getting different answers. That is very interesting. Okay, enough time for question 11 four. Now solve like question 12. Sir, if you're finished, both of which questions should we do? Then go to question 13 and solve first question of 13. But don't post the answers. Post the answers only when I tell you, fine? Okay, a lot of answers are coming. Now, I want to give answer to this question for this is a very good question. And the question is something like this. So it's seven plus five root three divided by seven minus seven plus root five divided by seven minus root five negative of seven minus root five divided by seven plus root five. One way to solve this question is that you take you rationalize the denominator here and then you start solving the question. Now, once you rationalize the denominator here, please try to understand you take seven plus root five divided by seven minus root five. So you have seven plus root five here divided by seven minus root five here. So this gives you nothing but seven plus root five whole square. So seven plus root five whole square is nothing but seven square is 49 root five square is five plus two into seven into root five, which is 14 root five. So that is nothing but 54 plus 14 root five. And in the denominator, you will get 49 minus five. That is equal to what 44. So this gives me nothing but 40. So the first part is this next part is seven minus root five divided by seven plus root five multiplied by seven minus root five divided by seven minus root five. So this is seven minus root five square. So this will give you 49 plus five minus 14 root five. Let me write on the other side here. 49 plus five minus 14 root five divided by 44. So this is nothing but 54 minus 14 root five divided by 44. Now, what happens? I have to subtract this when I subtract this. So I get here 54 plus 14 root five by 44 and minus 54 here. Negative positive will become negative this due to this negative and this negative will become positive. So I write plus 14 root five and everything divided by 44 because denominator is same. So this 54, 54 gone. So I get 28 root five divided by 44. I divided by four, I get seven by 11 root five. Now, when you compare here, there is no rational part. So A would be equal to zero. And in irrational part, it is seven by 11 root five. So B would be equal to one. Now, my question is, do you have to do these things separately? No, you don't have to do these things separately. Actually, if you look at the denominator, if you find the LCM of the denominator, actually you can solve the question in much shorter way. So LCM of the denominator is seven minus root five multiplied by seven plus root five, which is nothing but 45, 49 minus five cross multiplication will give you seven plus root five whole square minus, this will give you seven minus root five whole square. Now, I know that A plus B whole square minus A minus B whole square is nothing but four AB. So this can be written as four into seven into root five divided by 44, which gives me nothing but seven by 11 root five. So this takes only three steps to solve the question. Now, if it is seven by 11 root five, then my value of A comes out to be zero and value of B comes out to be one. So I think this method is actually very convenient to solve rather than the previous method. So if rationalization is at most of utmost importance, you can take the first method. But in any competitive exam, if this kind of question is coming, suppose in front of you in NTSC or somewhere else, you can utilize this method and find out the answer. So it's very necessary that you can find out the answer with the... You know the methods by which you can find out the answer. Two, three methods, if you know, that is always very, very beneficial. Now, let me solve the second question, which is question number 12. So A is equal to two plus root three. Let me find out one by A. One by A is one by two plus root three. And then I multiply it with two minus root three here and two minus root three here. So this gives me nothing but two minus root three divided by four minus three, which is equivalent to two minus root three. Now I have two plus root three minus two minus root three, so which is equal to two plus root three minus two plus root three. So two, two gone. This gives me two root three. So that's the answer. So two root three is the answer here. Now let me solve a few more questions. And those questions, let me select a few questions from here. And I think at the last, there are few long answer type questions. So why don't you solve question number seven first? And then we will solve one or two more questions from here, long answer type questions. So first solve question number seven. Let me know once you are done with the questions. Okay, I've started getting a few answers now. All answers are coming out to be seen out here. So let me check if anyone is getting any other answer. Few more answers now. Okay, so I'm getting few answers. I've got four answers till now. Two more minutes for this question. Though it could have been solved till now, but still I'm giving you two more minutes. Two more answers I've got. It's coming out to be same. Okay, let me just solve these questions. It's already enough time for this. Before I solve this question, I want to teach you a few concepts about this question. So what is happening here is, suppose I have 216 to the power, whatever, two by three. And don't take it from the perspective of question, just try to understand the concept here. Now, if it is 216 to the power, two to the power three. Now I the basic objective of solving this question is to remove this two by three. This is the main concept that how can I remove two by three? And by removing two by three, I won't have any problem if numerator is there. I have problem till the time there is a denominator in the power because we are not actually comfortable with powers having denominator. So I have to write six in such a format which has a power three or multiple of three so that this denominator three can be canceled out. What I'm saying is it is a to the power p by q. Now, a has to be written in some format where it is converted into c to the power q multiplied by p by q so that this q canceled out or d to the power n into q, multiple of q, so that again q can be canceled out. So basic objective of solving these questions is to cancel out the denominator in the powers given and then simplify these questions. So having told you these things, now what I want to do is I want to write it as first of all everything is negative so it goes in the numerator. So 216 to the power two by three plus 256 to the power three by four plus two into 243 to the power one by five. Now, I know 216 is six cubed. 256 is nothing but two to the power eight but here four is there. So two to the power eight will also do or I can write two square to the power four which is nothing but four to the power four. So this can be written as four to the power four. 243 is nothing but three to the power five. So you write four into six cube multiplied by two by three plus four to the power four three by four plus two into three to the power five to the power one by five. So this three and this three gone. I have six square 36. So I have four into 36 plus four to this four and four gone. So four cube is 64 plus this five and five gone. So two into three is how much six. So I have 144 plus 70. 144 plus 70 is nothing but 214. That's my answer. Now I would like you guys to solve because I started the class at 8.55. I would take it till 9.5. I quickly I would like you to solve this question five and question three. Quickly take two minutes each to solve these questions and give me the answer. Okay. I started getting answers. Okay. Aditya has given me some answer. Then Adith has given me some answer. Similarly, Nikhil has given me some answers. So good. Okay. Let me solve these questions but I have not got answers. Okay. I'll give you two more minutes to solve these questions. No problem. Solve it. Take your time. I've got three more answers. Ninth one is, where is the ninth one? I have given you question number five and three. Vashnav. Pushak is saying fifth one is 10. Maybe I'll have to solve these things. I don't have ready made answers. So I'm happy that most of you are giving me answers now. What I'll do is definitely I like to solve this question. Now I want to discuss this question. Question number five and then I want to give you an answer. So question number five is nothing but oh you mean the fifth one or no issues. No issues Vashnav. Good. I'm just doing a question. Just look at the question now. X is three plus sorry root three plus root two divided by root three minus root two and Y is root three minus root two divided by root three plus root two. I have to find out the value of X square plus Y square. So there are two options. I can directly find the value of X square. For that what I'll have to do is for that I'll have to take the square of this three plus root two divided by three minus root two and start taking the square. So I take square of numerator then I take square of denominator. The problem is this is root three. I'm sorry. The problem is I if you look at these numbers properly if you take it take the conjugate or sorry or if you rationalize this denominator. So if you rationalize this denominator you get root three plus root two in the denominator. And once you multiply it you get three minus two which is equal to one. So what I'll do is I will rationalize it first because once I rationalize it in the denominator one comes and then whatever is numerator I'll square it and add it over here. So X is equal to root three plus root two divided by root three minus root two and I multiply it with root three plus root two and here I multiply with root two. So this is root three plus root two whole square divided by root three minus root two multiplied by root three plus root two three minus two. So this gives me three plus two plus two root six divided by one. So which gives me five plus two root six and this is the value of X. What would be value of Y value of Y would be equivalent to just look at here. Value of Y is root three minus root two divided by the root three plus root two. So I multiply it with the root three minus root two and this is root three minus root two denominator is one. So I'm not writing this is root three minus root two whole square. So it gives me nothing but three plus two minus two root six. So this is five minus two root six. Now I have to take square of this. So this is X square would be equal to 25 plus two root six is four into six 24 plus five into two into two into five into two is 20 root six and Y square is similarly 25 plus 24 minus 20 root six. So if you add it, I get 50 plus 48 25 plus 25 is 50 24 plus 24 is 48 and this gets cancelled out. So my answer here is 98. Now let me move to the other question. The other question is I will rationalize it. So the other question is four divided by three root three minus two root two. So the other question is four divided by three root three minus two root two plus the other portion is three divided by three root three plus two root two three divided by three root three plus two root two. So this is nothing but 12 root three plus eight root two sorry root two and this is nine root three minus six root two. Why I'm doing by this method because already both positive and negative are available in the denominator which will give me a rational value in that denominator. So three root three square is nine into three twenty seven. Two root two square is nothing but what do you get eight? So you get 21 12 plus nine is 21 root three minus two root two eight minus six is two divided by 27 minus eight is 19. So I now keep the values. So 21 multiplied by 1.732 minus two into 1.414 I think 414 only has been given yes and divided by 19. So 21 into two is 42. So four is carry forward 21 into three is 63 plus four is 67. So six 21 into seven is 147 plus six 153. So three 15 and 21 plus 15 is 36. So this gives me 36.372 minus this gives me two point eight two eight divided by 19. Now when I subtract this I get 12 minus eight is four six minus two is four. Here I have three. So 13 minus eight is five and then I have five five minus two is three and then I have three here. So 33.544 divided by 19. So 19 into one is 19 and then I have 14 here. So 145 and 145 it goes how many times it goes 19 into seven is 133. So 145 12124. So it goes six times. So 114 104 and it goes five times. So 9590. So it is nearly equivalent to 1.765. So I hope that you are understanding this. One second. So what are what answers you guys are giving me? I got 1.765 rest all not getting any answer. So this should be the value. I think so I hope you enjoyed the class. Sorry. Yes, I'll set the value. The value that you have sent is how much two point two point zero six three. No, no, that's not right. I think I think because because because for me in the end it was it came out like 39.2 by 19. Okay, let me check once maybe you are right. Let me and I've got 36.372 plus 2.828. Yeah, actually you are right. There is a calculation mistake here. This has to be eight minus six is positive. So this is positive. So actually we need to add it. So how much did you get there? I got 39.2. This is 39.2. Yes, it will be two plus eight is zero. So zero and three plus eight is 11 12 39.2. So this gives me 2.063. Right? Yes, that's the answer. No issues. Some calculation mistake, which is my habit from long time. So I have it has costed me a lot of things in life. So you should not do calculation mistakes. Fine. Yes, sir. Okay, good. Who supported me with this? Aditya Kumar. Which school do you come Aditya? NPS, YPR. Oh, YPR. Good, Aditya. So thank you. And I hope you all enjoyed the class. There was some benefit from the class and wish you all the best. There is a test tomorrow from 9 to 10.30. You all know about it? Yes, sir. Okay, fine. So wish you all the best for that test. Give your best, use your best potential there to apply best of the concepts possible and then then get best of the results. Thank you so much for joining the class and wish you all the best for tomorrow's test. Thank you. Thank you, sir. Thank you, sir. Thank you, sir. Thank you, Aditya. Thank you for correcting. You're welcome, sir. Thank you, sir. Bye-bye.