 Hey friends welcome once again on or to this problem-solving session So we have been studying some theory part for some time now So as I told you in the last session as well You have to do a little bit of problem solving to get accustomed to the rules and the laws of exponents as it is going to be a very vital topic for your entire pursuit in Let's say education or in science. So hence if you are thorough with the rules and the laws It will be very easy for you to Understand and later on take up any higher studies in science and mathematics. So hence it is a very very vital Topic so pay at most attention and try to solve as many problems as possible So let us begin. So the first question is you have to simplify all these So 3 by 7 whole square into 7 by 2 to the power 4 into 2 by 3 square So the first law and I will be Writing the laws which I'll be using as I'll be using them. So first law, which I'm going to use is a by b Whole to the power n will be equal to a to the power n and b to the power n Isn't it so by this law this becomes 3 to the power 2 by 7 to the power 2 and then this is 7 to the power 4 and 2 to the power 4 and Finally 2 to the power 2 and 3 to the power 2, right? So let me rewrite all the bases together and so it will become 3 square by 3 square multiplied by 7 to the power 4 by 7 squared and This is 2 squared divided by 2 to the power 4 Isn't it now hence next one which I'm going to use is is this a to the power n Or m divided by a to the power m is equal to a to the power m Minus n. Okay. So hence this is 3 to the power 2 minus 2 into 7 to the power 4 minus 2 and This is 2 to the power 2 minus 4 Is it so let us see what are the values so 3 to the power 0 and This is 7 to the power 2 and this is 2 to the power minus 2 Now we'll use the law that 8 was 0 is always one Where a is not equal to 0. So this is always one then 7 Okay, so the next is a to the power m by n is equal to a to the power m minus n which we just saw and Again, another one is a to the power minus one or a to the power minus n will be equal to 1 upon a to the power n So using all this I'll get 1 into 49 Is it 7 square is 49 and this is 1 upon 2 square which is nothing but 49 upon 4 so I'm leaving it in fraction form. This is The first example next let's go to the second one and this question again It is similar question, but some exponents are negative over here and the base also are negative So I'll again do the same thing. I will first open the brackets up. So this is minus 2 whole to the power 3 and Divide by 3 whole to the power 3 into 5 to the power of minus 1 Divide by 6 to the power of minus 1 then it is 5 cube and this is 2 cube Right and the formula or the law I'm using is a by b whole to the power n is equal to a to the power n divided by b to power n. Okay now I'll club and Open also so minus 2 to the power 3 is nothing but minus 8. You can simplify here itself Then this is 3 to the power 3 which is 27, but I'm writing 3 to power 3 and So you can simplify this here a cell. So let us say this is 27 This is 27 then 5 to the power 1 will become minus 1 will become 1 to the power 5 And 6 to the power minus 1 will become 1 upon 1 by 6. So hence it will be nothing but 6 right. What do I do? What did I do? I did this 1 to the power 6 to the power minus 1 is 1 upon 1 upon 6 which is equal to 6 Isn't it now? This is nothing but 1 25 And this is nothing but 8 right So hence this 8 and this 8 will get cancelled and what else 3 2 ja and 3 9s are 27 right now 5 and this is 25 Okay, so hence the solution is nothing but negative sign was there. So negative will still be there 2 into 25 is 50 and in the denominated is 9 So hence answer is minus 50 upon 9 once again now in this question if you see It is again 3 bases are same and there are powers. So hence what can I do? I can write this as 3 times 3 to the power 1 not 1 plus 2 minus 3 to the power 9 p 7 plus 6 And what is the rule I'm using a to the power m Into a to the power n is equal to a to the power m Plus n okay So this will be nothing but 3 to the power 1 0 3 And this will be minus 3 to the power again 1 0 3 So both are same that means the difference will be 0 So this is the solution for this problem now in this problem It's given that a is equal to 3 b equals to 2 and then we have to find out the value of a to the power a Plus b to the power b. So let us find out a to the power a so I am solving a first So a to the power a will be 3 to the power 3 plus 2 to the power 2 So hence 3 to the power 3 is 27 plus 2 to the power 2 is 4 Okay, so hence the answer is 31 For b. Let us see. What is it? So a so I have to write 3 by 2 to the power 3 and it is 2 by 3 to the power 2 y b is 2 a is 3 So let us simplify this. So this is 3 to the power 3 by 2 to the power 3 into 2 to the power 2 and divided by 3 to the power 2 which rule a to the power m a by b to the power m or n is equal to a to the power m by b to the power m Okay, so hence now you club all the 3s together. So 3 to the power 3 by 3 square into 2 square by 2 to the power 3 which is equal to 3 to the power 3 minus 2 divided by or multiplied by 2 to the power 2 minus 3 which rule did I use? I used a to the power m by a to the power n is equal to a to the power m minus n Okay, so finally you'll get 3 to the power 1 into 2 to the power minus 1 which is nothing but 3 by 2 y because 2 to the power minus 1 is 1 upon 2 which rule a to the power minus 1 is equal to 1 upon a So using this we saw that this is 3 by 2 the other way to solve this could have been this that a by b to the power a and I could have written this as a by b to the power minus 1 to the power b Isn't it why? because b by a is equal to 1 upon a by b to the power minus 1 similarly a by b to the power minus 1 will be equal to 1 upon a by b which is equal to b upon a isn't it? So I could have written like this as well So hence now it will have become a by b to the power a Into a by b to the power minus b. So hence now the base is same for both So I could have written a to the power b to the power a minus b Isn't it? So now if you deploy the values you will get it directly. So a was 3 b was 2 Hold to the power 3 minus 2 which is 1 so hence 3 by 2 are the answer so same as before Okay, so there will be multiple ways to solve a same same solve the same problem So you can apply any method to solve these problems