 Hello friends welcome again to another session on mathematics and continuing with our series of lectures today we are going to start a new series and it is called SIRT. So SIRT is going to be a very important and integral part of your journey in mathematics all through your mathematical career. Okay so it is very important to understand what exactly it is and you know what are the different applications and how to do multiple operations on SIRT okay. So let's start with the definition. So what's the definition of a SIRT? A SIRT is arithmetical number or any root of so it says any root of any arithmetical number which cannot be exactly determined. Okay this is what the definition is. Now what does this mean? It means that so let us say root of two. Root of two is definitely a real number but not a rational number but root two cannot be expressed in terms of a recurring decimal right or a terminating decimal. So this decimal is non-terminating decimal representation non-terminating as well as non-repeating. So it's basically a irrational number right. So all such roots of any number which cannot be exactly determined is called a SIRT. Okay so examples are given. Root of seven you cannot really find out what is the value of root seven. Similarly root of eight, fifth root of four, fourth root of seven and root of 7.5 all these are examples of SIRTs. Okay so all these are examples of SIRTs. Now so you can also have SIRTs like root of x typically root of x root of y these are also considered to be SIRTs because you don't know what is the value of x and y. It may end up being a rational number but typically we consider root of x root of y or root of any such variable as a SIRT right. So what I mean is let us say if x is four then obviously it's not a SIRT. It's values two which is a rational number then in that case in one unique case root of x will not be considered as a SIRT but for you know for our you know usual purposes we consider root x root y or root of x plus y or root of x y all these are examples of SIRTs. Okay now now first property is a SIRT may sometimes be expressed as the product of a rational number and another SIRT. Okay so if you have a SIRT if you have a SIRT this can be expressed as a rational number let us say p by q something a rational number and multiplied by another SIRT. Okay so this is what is the first property we are going to learn. So let's see an example so this type of SIRTs for example root of 12 there is nothing outside root it's called a complete SIRT or a pure SIRT. Okay now root 12 if you see can be expressed as under root 2 square into 3 12 is 2 square into 3 so hence using the laws of exponents what we have learned 2 square into 3 whole to the power half will be the next step and then you can separate the powers like this and eventually you'll get 2 to the power 2 into half is 2 and root 3 is like this right so if you see there is a rational factor and there is a irrational factor isn't it so this one is irrational part and this one is rational so a complete SIRT that was a pure SIRT was converted into a product of a rational and an irrational factor but it it will not happen every time it happens only sometimes right how for example look at root 13 13 is a prime number so it cannot be expressed as a product of a rational and an irrational SIRT. Similarly root 15 is root 3 into root 5 again you cannot express it as a rational and an irrational SIRT rational factor and an irrational SIRT but examples like root 18 you can express is that it has 3 root 2 root of 24 can be expressed as 2 root 6 root of 50 can be expressed as 5th root 2 okay here there could be another examples also for example if I have 4th root of let us say 32 okay 4th root of 32 can be expressed as 2 to the power 4 into 2 whole to the power 1 upon 4 sorry I should not be putting this root sign it is 1 upon 4 1 upon 4 isn't it 1 upon 4 the same thing 32 can be expressed as 2 to the power 4 into 2 and whole to the power 1 by 4 so hence if you see it is nothing but 2 to the power 4 into 1 by 4 into 2 to the power 1 by 4 and what a law I am using a b to the power m is equal to a to the power m times b to the power m correct so hence what will this be this will be nothing but 2 into 4th root of 2 which can be expressed as this right so here also you see you can you express this as a rational factor and an irrational third right so this is one property second property is you can express a rational factor and a third as a complete sir the reverse of whatever we did the in the earlier part right for example if you have 5 root 3 you can express this as root of 5 square into root 3 and hence combining the roots you'll get root of 75 right so 5 root 3 is root 75 similarly 3 root 2 what is 3 root 2 root of 3 square is 3 into root 2 now since it is into so I can club the roots together and I'll get 18 mind you if it is it was root 3 square plus root 2 this will not be equal to root of 9 into 2 or rather even not equal to root of 9 plus 2 so be very very careful only when there is a into sign you can club the roots and you get this value so okay 7 root 3 is under root 7 square into root 3 and hence 49 times 3 which is 147 root right so you can take multiple such examples and see how you can express a rational factor and a third as a complete sir now this is an exercise which will be attached with this session so you can try them out so and see whether you are able to solve all these problems so try as many as possible and more than the number of problems you will solve more you will be comfortable with these operations