 to what is law number 3? Okay, there is this law. Okay, so far we have seen the base was same. Now this law where we are writing a to the power m whole to the power n is equal to a to the power mn. Okay, this is the again here a is a real number, real number and mnn are positive, positive integers, positive integers. Okay, so, so what is it? So how do we prove it? So let's prove, proof, how do you prove? Let's give us give them a proof. So am to the power n can be written as what a to the power m times a to the power m times a to the power m times how many such a to the power ms n times. Okay, which is nothing but a into a into a m times and same structure repeated how many times n times, right? So this entire thing goes on for how many times n times if you count how many a's are there in the first bracket m is how many here m is how many here and in all all such brackets m and how many such m's are there n m's are there. So total number of a is m into n. So hence it is a times m into n. Okay, okay, so because this is a into a into a how many times mn times, isn't it? Now interestingly if you see a to the power mn can be also written as written as a and m. Yep, because multiplication is commutative. So m into n is equal to n into m. So hence the same thing can be written as a to the power n to the power m, right? So what do we learn a to the power m whole to the power n is equal to a to the power n whole to the power m is equal to a to the power mn, right? Example, so let us take example 2 to the power 3 to the power 2 is nothing but what 2 to the power 6, right? So you can calculate also 2 to the power 3 is 8 and 8 square is 64. And this also 2 into 2 into 2 into 2 into 2 into 2 which also will also give you 64. So both are same. Both are same. So it works, isn't it? Similarly, 3 to the power 2 to the power, let's say 1 is nothing but 3 to the power 2 into 1 which is 3 to the power 2 which is 9, okay? Minus 2 to the power 2 whole to the power 4 is equal to minus 2 to the power 8. So this is equal to if you see this is nothing but 4 to the power 4. And here also it is minus 2 to the power 8 is nothing but minus 2 into minus 2 into so many times, how many times? 8 times. And in both the cases you will get 256 as the answer, 256 as the answer, right? This is law number 3. What is law number 4? So let us say law number 4, okay? Law number 4 says that if a and b are real numbers, real numbers that is they can be 0, the 1, 2, minus 3, minus root 2, whatever and m and n are m and n are positive integers, positive integers. Then what? Then 1 is a b to the power n is equal to a n times b to the power n. And second is a by b whole to the power n is equal to a to the power n divided by b to the power n, right? And b certainly cannot be 0 because then the operation becomes undefined in this case, in the second case, okay? You can again easily prove it a to a b to the power n can be written as a b into a b into a b, how many times? n times n times. That means a into a into a into n is multiplied by b into b into b, how many times? n times again. So hence it is a to the power n into b to the power n, is it it? Similarly, here also you can see this can be written as a by b times a by b times a by b, how many times? n times. So hence it is in the numerator a into a into a, how many times? n times divided by b into b into b, how many times? n times, right? So it is nothing but a to the power n by b to the power n. Take an example, let's take an example. Example could be 2 into 3 to the power 2 should be equal to 2 square into 3 square. Let's check LHS is nothing but 2 into 3 is what? 6 square, which is 36. RHS 2 square into 3 square is 4 into 9, which is also 36. Very good, right? Let's take another example. Example 2 by 5 let us say to the power 3 should be equal to 2 cube by 5 cube, okay? So 2 by 5 if you see, 2 by 5 is nothing but 0.4. So 0.4 cube must be equal to 8 upon 125 and actually if you see both will give you 0.064 as the answer. Now this is 0.064, this is the LHS and here also if you see 8 by 125 is 0.064. So both are correct, isn't it? So hence we saw there are four laws with integral exponentialized. So we will also see laws of exponents with non integral that is with fraction coefficients, sorry fraction exponents as well. So after this session we will be doing some problem solving sessions. Do go through those sessions so to understand these laws in a better way. Thank you.