 Hey, hello friends. So welcome again to another session on Serds and today We are going to discuss a very critical and very very important topic. That's called rationalization Now many students we have observed they struggle a lot when this particular concept is applied into Serds So let's try and understand these concepts in the best possible way and further to this you'll have to solve obviously Some amount of problems to get it affected. So let's begin So as the name suggests rationalization, that means you are converting something which is not rational into Something which is rational. That's why the name rationalization Now usually it is used for you know in context of Serds. So let us say If I have a Serd, let's say root 3. Okay, and I want to rationalize it So there exists another Serd which when multiplied with root 3 will render it a rational number So for example, if I multiply root 3 by another root 3 if you see what will you get you will get 3 Isn't it? Now 3 is a rational number. So what do we conclude? What do we conclude that this process of converting an irrational number Into a rational number by multiplication with another Serd is called the process of Rationalization you understood rationalization rationalization So I converted some irrational into Rational hence rationalization, right? So what do we understand or what do we conclude for any Serd? Whether it is a pure Serd or a compound Serd there exists another Serd Which when multiplied with the original Serd? Will give you a rational result? So let us take another example. Let us say I have root 2. I want to convert into a rational number So if I multiply either by root 2 or any multiples of root 2, I will get 2 So root 2 into let's say 2 root 2 will fetch 4 so everywhere you see whether you multiplied by root 2 or By 2 root 2 the result is always a rational number Rational number isn't it? So that's this process is called rationalization to convert a Ugly looking Serd into a beautiful looking rational number. So let us take an example of a compound Serd So let us say root 2 Plus root 3 is a compound Serd. I want to rationalize it What do I do? I multiply this with another Serd Let us say root 2 minus root 3 and why do I do this? You'll get to know in a short time. So if you now Multiply this it will be nothing but root 2 squared minus root 3 squared Which is 2 minus 3 hence minus 1 which is again a rational number Now this particular Serd which was multiplied with the original Serd. So this one is called the conjugate Serd What is it called? conjugate Serd Conjugate Serd, right? This is conjugate Serd or the another name is Complementary Serd Complementary Serd so these are the two names given to let us say Something which is be being multiplied with a compound. So this is the this is the compound Serd original compound Serd And you wanted to find out what should be it what should the other Serd be? Which will multiplied with this compound Serd will fetch you a rational number. So hence This is the Serd which is being multiplied is called the conjugate Serd, right? So every compound Serd will have a conjugate Serd. So what how do I define conjugate Serd? So we say so let us define this as well so that you get an understanding so to quadratic or to binomial binomial quadratic Now, you know make sure that the order is to so hence it is to binomial quadratic Serds Serds which differ Which differ? Which differ only? Which differ only in the sign only in the sign Which connects them? which connects them Right are called conjugate Serds, right? So example of conjugate Serds are Examples are for example example of root 2 plus root 7 What is the connecting sign connecting sign is this plus isn't it so root 2 plus root 7 conjugate will be nothing but root 2 Minus root 7 you can also say that root 2 plus root 7 conjugate could be minus root 2 Plus root 7 why because this could have been written as root 7 Plus root 2 so this is the conjugate of this and this is the conjugate of this, right? So these are called conjugate Serds. Obviously, you have to see that they are quadratic Quadratic order 2 so hence you can't say root through 3 Sorry third root of 4 Minus third root of 7 the conjugate is not third root of 4 Plus third root of 7. This is not going to yield you any rational number when you multiply these two So hence always remember this has to be for quadratic what Quadratic Serds, right? So that is how we define conjugate. So now, I hope you understood rationalization So what to do if there is a pure Serds, so you know this let us say if it is a pure Serd then Then what do you need to do? Multiply be multiply multiply it with itself and You will get rational Result, right? If it is a conjugate Serd, sorry if it is a compound Serd compound Serd Then multiply with its Conjugate, isn't it or complementary? Sir Example of one, let us say let us say example of one. So if it is root 3, I will multiply it by root 3 to get 3 If it was let's say root of 19, what do you need to do? Multiply by root of 19. You will get 19 right and examples of 2 examples of 2 if it is root 3 plus root 7 Then you multiply this with root 3 minus root 7 If it is 2 root 2 minus 3 root 5 Then you multiply this with 2 root 2 plus 3 root 5 if it is minus 2 root 7 plus 4 root 3 you multiply this with minus 2 root 7 minus 4 root 3 like that So these are all examples of you see everywhere. I'm just changing the sign. This is plus it becomes minus here This is minus this becomes plus here. So connecting sign Change, right? This is what we learned about conjugate Oh, sorry rationalization of Serds. Let us take an example. So example is rationalize the denominator rationalize rationalize the denominator Okay, so let us say first example is 1 upon root 2. I have to rationalize its denominator What do I know if it is a pure shirt multiply it with pure shirt? Same number, but then when you are rationalizing the denominator, you are changing the denominator basically So you have to multiply the numerator also with the same quantity to compensate the change So hence it is nothing but root 2 upon 2 now if you see the denominator has been rationalized Another example rationalizing 1 minus root 3 1 by root 3, sorry So hence it is nothing but minus 1. So you have to multiply with the same Sir same sir So hence it is minus root 3 by 3 rationalized now if it is a Let's say compound sir or a rational plus irrational sir. Let's say 2 plus root 3 So what do you do? You simply do 1 by 2 plus root 3 times 2 minus root 3 I change the connecting sign But I have to compensate also by multiplying the numerator so that the value doesn't change So what do I get I get 2 minus root 3 upon? 2 square minus root 3 squared now This is simply this one if you observe this is a plus b times a minus b form Which is nothing but a square minus b square so hence what do we get I get 2 minus root 3 upon 4 minus 3 hence it is 2 minus root 3 upon 1 that is 2 minus root 3 Let us take another Example so you have to let's say you have to rationalize 3 by 4 minus root 3 Yep, or let us say 2 root 3 right 4 minus 2 root 3 So how do I rationalize the denominator that is so it is 4 minus 2 root 3 multiplied by 4 plus 2 root 3 I multiplied with the conjugate and On top also it will be 4 plus 2 root 3 right so what should be the value now the value is 12 Plus 6 root 3 so I multiplied the numerator now and in the denominator you'll get 4 square minus 2 root 3 square again a plus b a minus b form so it is 12 plus 6 root 3 and Divided by 16 minus 12 right so you'll get 4 come sorry not 4 is not common so hence you'll get what 12 plus 6 root 3 divided by 4 and So you cancel can cancel 1 2 so it is 6 plus 3 root 2 3 root 3 sorry divided by 2 Okay, or you can further simplify it as 3 plus 3 by 2 Rule 3 so this is what? We get if we rationalize the denominator So likewise now you understand if it is a pure sword then you simply multiply by its No same number right but then since since we multiplied something in the denominator which will effectively change The value of the You know given you know sir, so hence what do I do? I multiply the numerator also to constant compensate that change and eventually I get this right so and if it is a Conjugate sir, then you know what to do Multi the sorry if it is a compound sir, then you know what to do multiplied by its conjugate top and bottom both to get The rationalized Combinator I hope you understood this and for more such you know you need to do some more Practice solve the worksheet which is attached and you will be able to perfect it