 Hello, folks. Welcome again to another session on Serds. So far, we have seen Serds, their definition, order of Serds, we compared Serds, we understood the meaning of complete Serds or pure Serds. We also understood how to add and subtract Serds and how do we used making the Serds with the same order and basically converting the Serds into the same order and then we understood how to add and subtract and compare. Now in this session, we will take up multiplication and division of Serds. So the process is very, very similar. So hence, let us take some examples and understand. So let us say we have to multiply third root of 6 with third root of 10. Now, if you notice here, the Serds are of the same order, isn't it? Both are third order. So when the Serds are of the same order, it becomes very easy to multiply. So what you can do is you can write this as 6 to the power 1 by 3 into 10 to the power 1 by 3. And then we can use the laws of exponents to write this as 6 into 10 to the power 1 by 3. And the law which I used is ab to the power m is equal to a to the power m into b to the power m. So now it is nothing but 60 to the power 1 by 3 or this is third root of 60. So in this case, in this example, the order was same. So hence, it was easier. Now let us say, let us take an example, whereas the Serds are of different order. So let us say I have to multiply fourth root of 5 with sixth root of 8. Clearly, this is fourth root. First one is, this is fourth order and this one is sixth order, isn't it? So how to do this? So again, in this case, we try to convert both of them into the same order. So 4 and 6, if you see the LCM is 12. So let us convert both of them into the order of 12. So 5 to the power 1 by 4 it is and this one is 8 to the power 1 by 6. Now 5 to the power 1 by 4 can be written as 5 to the power 3 by 12. We have to convert this into 12th order. And 8 to the power 1 by 6 can be written as 2 by 12. So what do I get? I get further 5 cube to the power 1 by 12 into 8 square to the power 1 by 12, isn't it? So now 1 by 12 is common and now we can repeat whatever we did in the example above. So this becomes 5 to the power 3 into 8 square to the power 1 by 12. So again, what rule? A, B to the power m is equal to A to the power m times B to the power m. So in the reverse order, we have used this law. So 5 cube by no is 125 and 8 square is 64. 8 square is 64 and this is whole to the power 1 by 12. So this will end up into 8000 to the power 1 by 12 or 12th root of 8000. This is multiplication of two serves of different order. What did we learn? Convert them, convert serves into same order, into same order and then multiply easily. Using the laws of exponents, simpler. Now let's take another example. So we saw two cases where serves are of the same order, serves of a different order. Here we will see serves of different order but within the root numbers are same. For example, third root of 2 multiplied by fifth root of 2. This one also is very, very simple. Why? Because it can be written as 2 to the power 1 by 3 into 2 to the power 1 by 5. So which is basically 2 to the power 1 by 3 plus 1 by 5 because we know A to the power m into A to the power n is equal to A to the power m plus n. So taking LCM n simplifying, you will get 2 to the power. 3 and 5 LCM is 15. So basically this will be 5 and this is 3. So hence it is 2 to the power 8 upon 15. Isn't it? Or we can say this is nothing but 15th root of 2 to the power 8 which is 256. So hence it is 15th root of 256. Now after seeing lots of multiplication, let's take up a division as well. So let us say I have to divide 6th root of 4 by 4th root of 6. This is the question. So in this case, what do we do? So it is nothing but we can write this as 4 to the power 1 by 6 divided by 4 6 to the power 1 by 4. Now if you see what is the LCM of 6 and 4? 12. So this can be written and reduced as 4 to the power 2 by 6, sorry 2 by 12, 4 to the power 2 by 12 divided by 6 to the power 3 by 12. Now we have, if you see the order of the two thirds are now same. So hence what can I say? I say 4 square to the power 1 upon 12 divided by 6 to the power 3 whole to the power 1 by 12 which can be further reduced as 4 square by 6 cube to the power 1 upon 12, isn't it? Why? Which rule? A by B whole to the power M is A to the power M by B to the power M. So hence it is 16 upon 216, 16 upon 216, isn't it? To the power 1 by 12. To the power 1 by 12. Now can we reduce it further? I think so, we can reduce it. Why? Because it is, yeah, so 16 is nothing but 8 into 2 and then 2 and basically this is 2 by 27, right? If you see this is 8 times 2 and this is 8 times 2 and 2 by 27 to the power 1 by 12. So hence the answer is 12th root of 2 upon 27. So this is how the division works. So we saw how multiplication works and if the order is same then you can simply multiply no problem. If the order is not same then first convert it into the same order and if the order is not same but the number underneath the root is same then also it is very simpler and then we also saw how to divide two thirds. So always remember making same order helps, okay? Thank you.