 Hello and welcome to the session. In this session we discussed the following question which says, find the sub-duplicate ratios of the following ratios. In the first part we are given the ratio x to the power 8 is to 25 y square. In the second part we are given the ratio 81 a square b square is to 36 a to the power 4 into b to the power 6. Before we move on to the solution let's discuss what would be the sub-duplicate ratio of the given ratio a is to b. Sub-duplicate ratio is given by root a is to root b or you can say a to the power 1 upon 2 is to b to the power 1 upon 2. This is the sub-duplicate ratio of a is to b. This is the key idea that we use for this question. Let's move on to the solution now. In the first part we have the ratio x to the power 8 is to 25 y square. We need to find the sub-duplicate ratio of this given ratio. Now the required sub-duplicate ratio is given by square root of x to the power 8 is to square root of 25 y square. Now since we know square root of x to the power 8 is given by x to the power 8 upon 2 which is x to the power 4. So we have x to the power 4 is to now square root of 25 y square is given by square root of 25 which is 5 and square root of y square which is y. So this is 5 y. So this is the required sub-duplicate ratio. Thus we have sub-duplicate ratio of x to the power 8 is to 25 y square is x to the power 4 is to 5 y. This is the answer for the first part of the question. Now next we have in the second part we are given the ratio 81 a square v square is to 36 a to the power 4 v to the power 6. The required sub-duplicate ratio is given by square root of 81 a square v square is to square root of 36 a to the power 4 v to the power 6. Now square root of 81 a square v square is given by square root of 81 which is 9 square root of a square that is a square root of v square that is v. So 9ab is the square root of 81 a square v square. Now square root of 36 a to the power 4 v to the power 6 is given as square root of 36 which is 6 square root of a to the power 4 that is a to the power 4 whole to the power 1 upon 2 into b to the power 6 whole to the power 1 upon 2. Thus we have 6 into a square into v cube. So 6 a square b cube is the square root of 36 a to the power 4 v to the power 6. So this is the required sub-duplicate ratio that is sub-duplicate ratio of 81 a square v square is to 36 a to the power 4 v to the power 6 is equal to 9ab is to 6 a square b cube. This is the answer for the second part of the question. So this completes the session. Hope you have understood the solution of this question.