 Hey folks welcome again to another session on exponents of real numbers now in this case We are going to study rational exponents of a real number. So, you know, what is a rational number? What is a real number? So, let us say we have a real number. We have a real number a Okay, a is a real number. So it can be expressed like this as well a belongs to the set of real numbers So a is a real number that means it could be anything like minus 1 1.5 even 0 even root 2 whatever right now. Let us say there is another Rational number. So, let us say there's a rational number P of the form P by Q. Okay, this is a rational number This is a rational Rational number. Okay. Now What is meant by a to the power P by Q? So, a to the power P by Q will be written as a to the power P to the power 1 by Q Okay, by the laws of exponents, we know so a to the power P whole to the power 1 by Q can also be written as Qth root of Qth root of a to the power P Okay, this is what is meant by rational exponents of a real number example So if we take an example, let us say a is 2.5 To the power, let us say 1 by 2 Okay, so it is nothing but 2.5 to the power 1 and this whole to the power 1 by 2 Okay, let's take another example 3.5 to the power 3 by 5 will be nothing but 3.5 to the power 3 and this whole To the power 1 by 5 Okay, this is what is meant by rational exponents of a real number Now that you have to take a little precaution here because let us take an example where it is negative 2 to the power, let's say 3 by 6 Okay, so will be nothing but It is will be nothing but by our definition negative 2 to the power 3 and this whole to the power 6, sorry 1 upon 6 Correct. So this is nothing but minus 8 to the power 1 by 6 now, you know that even root of any negative number is not real So you will never get any real real value for this. This is Not a real value. This is not not a real value. Why? Because if there is a 6th root of negative 8, then if we multiply it 6 times to itself It will never get you a negative number. What do I mean? Let us say negative 8 and the 6th root of negative 8 is let us say x This means what? x into x into x 6 times right 6 times must be equal to negative 8 Now any number which will multiply to itself 6 times will never give you a negative number Why? because let us say if x is positive then all x's are positive then you'll get a positive number and Let us say if x is negative then this negative into this negative will become positive This negative this negative will again become positive and this negative negative again becomes positive and three times positive three positive numbers when well-replied Will give you a positive number only so there will not be any case no case no case where where x to the power 6 is equal to minus 8 not at least in real number set yes, if you Go to your higher level in Mathematics, then you'll know that there is something called complex numbers or imaginary numbers Where this is possible, but otherwise this is simply not possible in real number set Okay, so hence while dealing with rational exponents We must be careful that we don't land up into something which is not defined in the Real number set otherwise all other rules stay as they are or as they as we discussed in the case of integral Exponents so now let's do a quick recap of the laws So here also the laws of exponents will be similar to whatever we discussed in the case of integer exponents now again, so if a belongs to the set of real number and let us say m and n are Rational numbers Okay rational Numbers then First law says a to the power m into a to the power n is equal to a to the power m Plus n Yeah, in the integral experiments also we saw the same law second law is a to the power m divided by a to the power n is Equal to a to the power m minus n So let's take one examples Parallel, so let us say I'm taking a as 3 so 3 to the power. Let's say 2.5 Into a to the power. Let us say 1.5 Well, sorry a is nothing but 3 here. So let us say a is 3 So this will be nothing but 3 to the power 2.5 Plus 1.5, which is equal to 3 to the power 4 Okay, so now 2.5 and 1.5 are rational numbers. You would have noticed by now similarly 3 to the power 2.5 divided by 3 to the power 1.5 is Nothing but 3 to the power 2.5 minus 1.5, which is equal to 3 to the power 1 Right, let's take the third law Third law of exponents is a to the power m whole to the power n is Equal to a to the power m times n and we saw the same thing in case of integral Exponents as well where m and n were Integers so hence now 3 to the power 2.5 Whole to the power 1.5 is equal to what 3 to the power 2.5 into 1.5 Okay, now fourth fourth law is a to the power minus n is nothing but 1 upon a to the power n Right, this means 3 to the power minus 2.5 is equal to 1 upon 3 to the power 2.5 Okay, so you know you're already you're already familiar with these laws So it should not be a big problem now a to the power m by n will be equal to a to the power m Whole to the power 1 by n the only thing you have to Take care is the nth root should be defined For the number you obtained or the same thing can be written as a to the power 1 by n Whole to the power m. Is it that so same thing can again be written as nth root of a to the power m or nth root of a Whole to the power m whichever way you want to Write it. So this is and this nth root is This is nth root of a Okay, similarly sixth root is A b now, let us say b also is a real number also, you know, and I'm writing it adding it here. So b is also a Real number then a to the power a b to the power m will be equal to a to the power m times B to the power m now this dot don't get confused with decimal. This is not decimal. This is multiplication sign here Okay. Now so example 2.5 into 1.5 times to the power, let's say 1.7 will be equal to 2.5 to the power 1.7 multiplied by 1.5 to the power 1.7 This is what it means. Now. There's one more law 7th law which is a by b whole to the power m is equal to a to the power m B to the power m. So let's take an example 2.5 by 1.5 whole to the power 1.7 is equal to 2.5 to the power 1.7 Divide by 1.5 to the power 1.7 Okay, so these are these are the laws of A laws of Exponents real x rational exponents of a real number So I would suggest you please take down these laws and write down in a piece of paper or maintain a diary of all the rules and formulae and then whenever you are solving problems you keep in front of You know, you keep you open that formula book and use it as much as possible while solving problems So actually don't need to mug up any formula