 Welcome again friends. So in this session, we are going to discuss integral exponents of a real number So what does it mean? So it means integral exponents So we learned in the previous section that a to the power b if it is given so b is your exponent, isn't it? exponent and a Happens to be your base. Okay, so base raised to the power exponent and we are trying to work or work On these kind of numbers now We are going to deal with a special case here where B is a B is an integer B is an integer Okay, so B is an integer like what I like. Let us say 3 0 minus 1 2 minus 7 and so on and so forth and B is not equal to numbers like 0.1 root 2 Etc. Okay, so B is an integer B is an integer now when B is an integer. There are three possibilities one is So one is when B is greater than 0 B is when B is greater than 0 This is case one when second is when B is equal to 0 and Third one is we are going to discuss when B is less than 0 Okay, now if B is greater than 0 then a can be any real number a can be Any real number any real number, right? So when B is greater than 0 a can be any real number, which is not true for this Now when B is less than 0 then we have to take some precautions. Why because by definition you will see B less than 0 case we have to be a little cautious in terms of In terms of a so a in this case cannot be 0 Okay, so we'll see once we define and B equals to 0 we again have a definition where a to the power B in this case That is a to the power 0 is defined as 1 okay, so please remember if B is greater than 0 a can be any real number if B is less than 0 then a cannot be 0. Okay, why because After some time and we'll discuss what exactly it means when B is less than 0 then it will become much clearer to you Now be greater than 0 so you know what to do if B is let us take this case case number one So in case number one B greater than 0 so a is any real number So a to the power B is nothing but a into a into a how many times we do it B times This is for a is equal a be greater than 0 and a is a real number Now for example 3 to the power 2 will be equal to 3 into 3 2 times 2.5 to the power 3 will be equal to 2.5 into 2.5 into 2.5 3 times Then let us say 4 by 7 to the power 4 will be equal to 4 by 7 into 4 by 7 into 4 by 7 into 4 by 7 Okay, now in the second case In the second case When B is equal to 0 right this case was be the first case was be greater than 0 Second case is B equals to 0 in this case by definition Any real number but 0 so a to the power of 0 will always be equal to 1 mind you a should not be equal to 0 in this case also why because 0 to the power 0 is not defined is not defined Okay, please remember 0 to the power 0 is not defined Is it it so this is not defined and So for for example 2.3 to the power 0 is 1 4 by 6 to the power 0 is equal to 1 Minus 2 to the power 0 is also equal to 1 right and third case is third case is what B is less than 0 again a in this case cannot be 0 a can be any real number but 0 why because by definition a by b if b is less than 0 is given by 1 upon a to the power Minus b right so hence if you see here a is in the denominator so the denominator Denominator cannot be 0 ever Cannot be 0 because dividing by 0 is not allowed Is not defined so hence example let us say 3 to the power minus 2 will be equal to nothing but 1 upon 3 to the power Minus of minus 2 which is nothing but 1 upon 3 Squire right minus of minus 2 c minus b. I have to put a minus sign in b Similarly 2.5 to the power. Let us say minus 7 is equal to 1 upon 2.5 to the power 7 okay, and let us say 5 point or 5 by 3 to the power minus 4 is equal to 1 upon 5 by 3 to the power 4 okay, so by Definition if you see by cross multiplying multiplication you can also see 2.5 To the power 7 can be also expressed as 1 upon 2.5 to the power minus 7 Correct here also if you see this is nothing but 5 upon 3 to the power 4 If you cross multiply you will get what 1 upon 5 upon 3 to the power Minus 4 so if there is a negative sign you want to make it positive You have to reciprocate the entire stuff. So let's take an example. So another example to show you So 2 to the power 3 into 2 to the power Minus 3 will be equal to what? 2 to the power 3 into 1 by 2 to the power 3 Is it and this will be equal to 1 Okay, so this is an example. So please keep in mind. So what did we learn? We learned that a to the power p is defined for positive integers b Sorry integers b Where b can be greater than zero b can be equal to zero and b can be less than zero as well But a in the last two cases that is when b is zero and less b is less than zero then a Cannot be zero. So please be careful about it Okay, so when b is greater than zero then if a is zero then is zero to the power anything Let us say seven is also zero. So here there is no problem But zero to the power zero is not defined. So please remember not defined not defined and one upon Zero to the power or rather zero to the power any negative number. Let us say Let us say zero to the power minus seven is not defined Not defined Right. Sorry is not defined. So please please take keep this in your mind in the next session, we will go ahead and understand other laws of exponents