 Hi friends, how are you guys? I hope you're enjoying mathematics now Continuing with our problem solving sessions. This is another session in which we are going to discuss These two questions. So it's given that x and y are positive real numbers Then you have to simplify under root x to the power of minus 4 y to the power of 5 This is one and the second one is this looks too intimidating, isn't it? But don't worry, you'll be able to solve it using the laws of exponent. So how to solve both of them Let's start with the first one first one says x to the power minus 4 and y to the power of 5. Now, I know most of you Get frightened or intimidated by seeing the variables, but don't worry, you know Working with variables is much easier than numbers because relatively Calculations are lesser. Okay. So hence How do I go about it? So I'll use the laws of exponent. So under root anything is nothing but x to the power of minus 4 y to the power of 5 and this whole to the power half, isn't it? Which law did I use? nth root of anything is Equal to a to the power of 1 by n. Now why a has been given as positive is we really don't want to take chances Where we have to find out that I say square root of a negative number. So hence to Avoid such situations. It is already given that x and y are positive. So no problems with negative Square, you know Let's say square rooting that number. Okay. Now, what is the next step? Next step is this can be written as x to the power of minus 4 half times y to the power of 5 to the power half which rule I'm using A b whole to the power m is nothing but a to the power m times b to the power m now again Mind you, this is not a decimal. This is a product Okay, so hence here What is next step next step is x to the power minus 4 into 1 by 2 Into y to the power 5 into 1 by 2. So what rule again? So a to the power m Whole to the power n is equal to a to the power m times n Okay, so please I'm writing these rules again and again. That's such that it gets Uh reinforced and you know, uh, it gets permanently set in your mind Now x to the power minus 4 into half no brainer It is x to the power of minus 2 and it is simply y to the power 5 by 2. Isn't it? So hence I can now write it as y to the power 5 by 2 Divide by x squared y Because a to the power minus n is equal to 8 1. Sorry not a 1 upon a to the power n So these are the rules which we used And now you see you cannot really reduce it further Okay, so hence this should be the final Answer now, let's take the Next question So here is the next question guys. So looks a little intimidating. It's complex looking question, but trust me While you do three fourth of such types and you'll be very very comfortable solving all these questions now Let us see how to let us see how to Solve this so hence the the step is nothing very very simple You take each terms separately and simplify to the maximum possible extent Now let us say First term root x right so hence I can write this as x to the power minus 2 by 3 whole to the power half right next y to the power 4 by 2 and then This divided by x y to the power minus half Whole to the power half. Isn't it? This is what it is right now. Okay. So next is Again, this will be nothing but x to the power minus 2 by 3 into 1 by 2 and the rule I'm writing here a to the power m to the power n is equal to a to the power mn Isn't it now? This is nothing but y to the power 2 simple arithmetic and then divided by divided by x to the power half and y to the power minus half to the power half Okay, now let's again keep simplifying. So it is x to the power minus 1 by 3 Why this 2 and this 2 will go? times y square divided by x to the power 1 by 2 And then it is y to the power minus 1 by 4. Isn't it? Why will that be because a to the power m again same formula a to the power m to the power n is a to the power mn Now, let us simplify further. So this will be y square divided by x to the power 1 by 3 And if you see this is nothing but into 1 upon so since it's a division so it is x to the power 1 by 2 and y to the power minus 1 by 4 Isn't it? Since it's division it will go into the denominator Now, let us simplify further. So if you see it is y square into y to the power 1 by 4 why because a to the power minus n Is equal to y to the power a sorry a to the power n Right, so hence or vice versa. You can say a to the power n is equal to 1 upon a to the power minus n So I adopted this rule to get y to the power 1 minus 1 4 1 by 4 in the numerator, okay Now divided by what is left for x x to the power 1 by 3 into x to the power 1 by 2 now Right. So what is this now simple y is the common base. So hence it will be 2 plus 1 by 4 divided by x to the power 1 by 3 plus 1 by 2 which rule did I use again a to the power m into a to the power m is equal to a to the power m plus n is it it so hence final answer y to the power take lcm so it is 8 plus 1 by 4 and here take lcm it is 2 plus 3 by 2 So hence finally you will get y to the power 9 upon 4 And 9 9 upon 4 divide by x to the power 5 by 5 by sorry, this will not be 2. This would be 6 here So 5 upon 6 right. So y to the power 9 by 4 divided by x to the power 5 by 6 This is the final answer. So again, please remember how to utilize the laws of exponents to solve such problems