 Hello friends once again, we are continuing our practicing practice session and This time we have taken this problem 25 to the power 3 by 2 into 243 to the power 3 by 5 Divide by 16 to the power 5 by 4 into 8 to the power 4 by 3 It's always a good practice to remember some powers of common, you know prime numbers and other such numbers For example, you must remember what is a 5 square which is 25 must also remember 5 cube Which is 125 5 to the power 4 is 625 and so on and so forth similarly powers of 2 you must remember so 2 is powers are 4 8 16 32 64 128 2 5 6 5 1 2 and 1 0 2 4 These are the 10 powers on 2 similarly powers of 3 are 3 then 9 then 27 then 81 then 243 Then 729 so these are the powers of 3 then similarly for 7 it is 7 49 then 343 Likewise right so these are some powers of 7 you must remember So that the calculation becomes you will be a little faster while doing the calculation So you can see 243 and here also there is a 243 so you could have figured it by now What is it going to be now how to solve so the method is? Prime factorize the number and then use the laws of exponents. So what is that method? factorize prime factorization so do prime factorization of such numbers prime factors you find out and then use laws of exponents Correct. This is how you have to do now. Let's begin so 25 can be written as 5 square This is the prime factorization and then the exponent is 3 by 2 Then multiplied by 243. What did we see 243 is nothing but? 2 3 to the power 5 actually 3 to the power 5 and then this is 3 by 5 as the power and then 16 can be written as 2 to the power 4 and the power is 5 by 4 and 8 can be written as 2 to the power 3 to the power 4 by 3 and then Simply using our laws of exponents. I know first term is 5 2 into 3 by 2 and The rule used is a to the power m whole to the power n is a to the power m times n and Then multiplied by 3 to the power 5 into 3 to the into 3 by 5 sorry the power is a 3 by 5 Then 2 to the power 4 into 5 by 4 Multiply by 2 to the power 3 into 4 upon 3 very simple. Isn't it now it just plain arithmetics, right? So 2 Goes 5 goes 4 goes and 3 also goes so hence I get 5 to the power 3 multiplied by 3 to the power 3 Divide by 2 to the power 5 and then divide by 2 to the power 4 Isn't it? This is what you will be getting now So that means it is 5 to the power 3 that is 125 See 5 to the power 3 was 125 isn't it 125 and 3 to the power 3 is 27 and this divided by 2 to the power 9 actually it is 2 to the power 5 into 2 to power 4 is 2 to the power 9 Which will be giving you 500 and Well, if you multiply the numerator here, you will get 3375 Divide by 512 so this should be the answer Okay, so what did we learn we learn in such plainly numeric question of Exponents try to first of all prime factorize the numbers and then use laws of exponents my dear friends, okay So in this problem again, if you see there are four terms numerator and denominator there and there is a minus sign which might Create some trouble because we haven't studied Any law which uses any law of exponents which uses or which is Defined for let's say two terms separate by plus or minus sign because if you remember all those were product based for Rules so it a into b to the power n is a to the power n b to the power n But we never studied anything like a to the power a plus b to the power n is what? Okay, so never mind. Let's see how we can solve such questions So if you see all are appearing to be powers of two, isn't it? So let us again going by our basic rule how to do first prime factor So 16 is 2 to the power 4 clearly and then this term is 2 to the power n plus 1 Minus 2 square is 4 and 2 to the power n. This is the numerator The denominator again 2 to the power 4 into 2 to the power n plus 2 Minus 2 into 2 to the power n plus 2, right? So what to do again? So hence basically if you see The you can use the law of exponents and you'll see 2 to the power it is n plus 5. Why? because 2 to the power m plus 2 into sorry into 2 to the power n will be 2 to the power m plus n Is it it? So I use this in the first and then the second one also it is 2 to the power n plus 2 divided by 2 to the power again n plus 6 Minus 2 to the power n plus 3 Isn't it? That's what we are getting now if you see closely This something can be cancelled why and how how to find that out So if you see there are 2 to the power n plus 5 Exponent and n plus 2 on the second term on the numerator Then I can write the first term as 2 to the power n plus 2 In 2 2 to the power 3 isn't it? Why because the sum of the exponents will give you n plus 5 Minus 2 n plus 2 right I can write like that and why did I do that because I got the hint from this n plus 2 I now want to in the next step. It will become much clearer now in the second case also I can write 2 n plus 3 into 2 to the power 3 Minus 2 n plus 3 Isn't it? Now if you see I can take 2 to the power n plus 2 common from the numerator And what is left within 2 to the power 3 and then minus 1 isn't it? And in the denominator as well I can take 2 to the power n plus 3 as common and Then common the rest factor is 2 to the power 3 minus 1 Isn't it now clearly this cancels out this so what is left? I am left with 2 to the power n plus 2 Divide by 2 to the power n plus 3 isn't it which can be written as 2 to the power n plus 2 n plus 2 minus n plus 3 Isn't it? Why because a to the power m divided by a to the power m will be a to the power m minus n Now hence what will be the result result would be 2 to the power n plus 2 minus n minus 3 Which will give me 2 to the power minus 1 which is finally 1 upon 2 Why because a to the power minus n is 1 upon a to the power n right So hence the answer is 1 by 2 of for this particular simplification problem