 Hello friends. How are you? I hope you are doing fine and you're enjoying math sessions. So continuing with our series today We are going to discuss principal and it's root of a positive and a negative number Okay, so in it through first of all, let us try and understand. What does and it mean in it Right when we say and it is nothing but so when you write first you write How do you express it? You write one and then you write ST as a superscript Similarly, let us say after say eight. So I'll write eight and th right so n stands for a positive integer and th stands for that Position for example first position second third fifth ninth ninety nine hundred like that. Okay, so nth we pronounce it and it Now root of a positive and a negative number. We have to understand now. Let us say n is a positive integer and A is a real number examples of real number, you know and that to positive real numbers We are talking about so examples are two two point five three point three three root two, right and so on and so forth There are infinitely many positive real numbers now examples of positive integer n is Let's say one two three ninety nine one million whatever right. So these are all integers Now we are talking about nth root of a Number, okay, so nth root of a we have taken a let us say that nth root is defined Or let us say the value is x then how do we define nth root? Very simple What do we say is what should be the power on x so that? You know, or let's say if I raise X to the power n then I must get a that's what is called nth root So x here becomes the nth root nth root that means if you raise x to power n you will get a Okay, that's what here it has been explained isn't it so nth root of any positive number is another number What kind of a number such that if that nth root if you raise it to power n You will get the desired number or let's say the number for which you were trying to find the roots out I hope I know, you know just by saying in words. It will be very difficult to understand But we'll sooner take examples as well. Also, please remember When x is the nth root of a we express s x as a to the power 1 by n Okay, if again, so we'll take examples. It will become much clearer So right now understand if x is nth root of a that is x to the power n is a Then x can be represented as a to the power 1 upon n So if you see the powers are getting reciprocated, isn't it? So whatever was power on x on a it is 1 upon that n isn't it? now this is this particular value is called nth root of a and it is expressed also as like this right so you simply take a Root sign like this and you write a if it is a square root Then you don't need to write anything and over here So if you see not usually the standard or the process is like this But when n is 2 you don't need to write 2 over there. You can simply write root of a so this is popularly known as root of a Third root of 64 for example, you see I have taken third root of 64 is nothing but 4 why because if I raise 4 to the power 3 you will get 64. So hence hence third root of 64 is 4 okay now 4th root of 81 what is 4th root of 81? So what should be that value which when raised to power 4 will give me 81 and that value is simply 3 Isn't it that value is simply 3 so this is if you raise 3 to the power of 4 you will get 81 So hence we say 4th root of 81 is 3 Okay, similar. So 3 is as I have written here 3 is a 4th root of 81 Why because if you raise 3 to the power of 4 you'll get 81 Okay, now similarly another example third root of 8 is 2 and 10th root of 1 0 2 4 is 2 again, right? Why because 2 to the power 10 is going to be 1 0 2 4. So hence 10th root of 1 0 2 4 is equal to Okay, now what happens when a is less than 0 guys, so let us see that a is less than 0 means a is a negative number now There is a problem. The problem is we can't have Yeah, we can't have Even Numbered root of a negative number. What do I mean? So I can't have second root fourth root sixth root eighth root like that offer negative number That's not defined in mathematics in the set of real numbers. So for example, if I ask you what is second root of minus 4 So by definition of second root is what x should I raised as to power 2 to get minus 4? Now x in x square is nothing but x into x 2 times is equal to Minus 4 that's what so you have to find out of x which when multiplied by x itself will get you minus 4 but tell me X can be either negative or positive So a negative when multiplied by negative will be positive and a positive and multiplied by positive will also be positive Guys, there will be no case where you multiply Two positive numbers and you get a negative number Similarly, you cannot have two negative numbers when multiplied you will get Negative number you understood. So hence There is a problem. We don't know what x should be there which when multiplied will yield minus 4 So hence second root of minus 4 is not defined in real numbers Yes, it is definitely defined in a set of numbers called imaginary numbers Which is again a subset of complex numbers, which you'll study later on similarly fourth root of anything will not be defined fourth root of Negative number not anything fourth root of let us say Minus 16 now if it be it were 16, then you know fourth root of 16 is 2 y fourth root of Fourth root of 16 is very clearly 2 y because 2 into 2 into 2 into 2 to the power 4 that is is 16 But tell me guys. You are multiplying 4 x is so 4x. Let's say Fourth root of minus 16 is let us say x. So that means you multiply x 4 times to get minus 16 now. Tell me X these are all same is it all are same value. So either all of them will be negative or all will be positive Now there are four positive numbers multiplied together getting you negative not possible and Four negative numbers getting you negative again is not possible. So hence my friends X doesn't exist into a number set right now So hence Likewise, you cannot have sixth root of negative number eight root of negative some number. Why because The moment it is even numbered you'll never get product of even number as in even number number of Numbers or that means what I mean is four four numbers Four same numbers six same numbers eight same numbers ten and so on and so forth If ten same numbers are multiplied you will never get a negative number. That's what I'm trying to explain So how do I do find? Even number root of negative number in real numbers. It's not defined But definitely n can be odd. Isn't it n can definitely be odd so if n is odd then nth root of a negative number is nothing but Take the minus sign out and find the nth root of mod of a what is mod mod is nothing But absolute value of a so example has been taken. Let us say you have to find out third root of minus 27 So what do you need to do take the minus sign out and find the absolute value of 27 and find the third root of that mod 27 minus 27 So the mod of minus 27 is simply 27 because mod is always positive Third root of 27 is 3 minus sign was there since third root is minus 3 Why now you can justify also minus 3 multiplied by itself 3 thrice you'll get minus 27 Similarly, what is fifth root of minus 32? So once again you take the minus sign out then take mod of minus 32 take the fifth root And you will get minus fifth root of 32, which is minus 2 why because minus 2 to the power 32 minus 2 to the power 5 is minus 32 Just for you know another example and if another example could be let us say third root of negative minus 125 Okay, so what do we do you simply take out the minus sign then find the Mod of 125 that is absolute value and find the third root of odd of 125. So what is this value minus 5? Isn't it? So that's how so and you also understood that odd numbered numbered root of a negative number negative number is Also negative. So you will not get a positive and it's root of a negative Number so please keep these things in mind. It will be useful in many many applications of mathematics. Thank you you