 Hello friends. So as you know problem solving is an integral part of any concept understanding So we are continuing with our series on problem-solving Involving surge. So the question here says rationalize the denominator. So mind the words It's rationalize the denominator, right? You can rationalize the numerator to But rationalizing the denominator helps in simplifying many search. Hence, we are saying rationalize the denominator and the Expression here is under root 1 plus x square minus under root 1 minus x square Divide by under root 1 plus x square plus under root 1 minus x squared You know how to rationalize a third and especially if it is a compound search If you see this is a compound third mix of two search, isn't it? So what do you do? You multiply it with it with It's conjugate So let's do that. So hence it is 1 plus x squared Minus under root 1 minus x squared will take the denominator and multiply it with its conjugate So this was the original denominator and now I am multiplying this with 1 plus x squared minus again 1 minus x squared and This whole again is 1 plus x squared under root minus Under root 1 minus x squared. So you have to if you are changing the denominator, you have to compensate the change by multiplying The numerator as well. So that's why I knew I multiply and divide the given expression by the same So and that's what is nothing but conjugate of the denominator Now you could have also done the same thing by multiplying with this conjugate So 1 minus x square minus 1 plus x square So you could have done this as well that means you could have changed the sign of this one And found the conjugate and multiplied answer would not be impacted Now let us go ahead and complete the multiplication process. So what is it? Let us multiply This with this first. So this is the multiplication. So let me draw with a different Color so that you know, you don't get confused. So this is first multiplication is this Right. So what do I get? I get Root of 1 plus x squared into root of 1 plus x squared, right Then what do I need to do? I need to multiply these two, which is nothing but under root 1 plus x squared times Uh, actually, this will be a minus sign here because I am multiplying a negative factor. So 1 plus x square times under root 1 minus x squared then what then minus under root 1 minus x squared times 1 plus x squared What am I doing? I am multiplying these two now And then finally, what do you get? You get minus minus plus under root 1 minus x squared times under root 1 minus x squared This is the numerator and the denominator, you know, this is a plus b a minus b form So hence you can write 1 plus x squared squared minus under root 1 minus x squared squared Right Look closely. This is a this is b. This is again. This is a plus b. This is a minus b. So hence it will be a square minus b square That's what I did now What is let's uh, simplify. So if you see the first term on the numerator, it is nothing but 1 plus x squared the root will go And the second term is nothing but under root 1 plus x squared times 1 minus x squared Same thing here minus under root in fact, you don't need to do this if you see Yeah, actually this will become two twice of this So hence minus same thing But we just did 1 plus x squared And then what this is nothing but 1 minus x squared when you square This term you'll get 1 minus x squared right divided by again 1 plus x squared then minus 1 minus x squared Right, this is what you will get next Let's simplify. So if you see this x square This x square will go and hence what will be left you will be left with two Minus this thing will become Two times because this item and this item are same. So two times this And what is within so you can write 1 minus x to the power 4 why? Because 1 plus x square times 1 minus x square. So a plus b a minus b will be equal to a square minus b square hence it is 1 square minus x to the power 4 Okay, so I hope you understood this process and divide by what? So 1 minus 1 will get cancelled you'll get 2 x squared So if you can eliminate all the tools you'll get 1 minus under root 1 minus x to the power 4 divided by x squared So hence if you see I have now got rid of any irrational component or factor in the denominator So the denominator has been rationalized. This is what the question demanded and hence We have obtained the right result. So this is what is called rationalizing the denominator Using conjugate surge