 Hello and welcome to the session. In this session we discussed the following question which says, express as a rational number 2 upon 3 whole q, this whole square divided by 2 upon 3 whole square multiplied by 1 upon 2 to the power minus 2, multiplied by 3 to the power minus 1, multiplied by 1 upon 6 whole to the power minus 1. Before proceeding with the solution, let's recall some facts for any given rational number a upon b where m and n are taken as positive integers. Then we have a upon b to the power m multiplied by a upon b to the power n is equal to a upon b to the power m plus n. Then a upon b to the power m divided by a upon b to the power n is equal to a upon b to the power m minus n when we are given m greater than n and this would be equal to 1 upon a upon b to the power n minus m when we are given n greater than m. Also a upon b to the power m and this whole to the power n is equal to a upon b to the power m n. Now let's move on to the solution. The expression given to us is 2 upon 3 whole q and this whole square divided by 2 upon 3 whole square multiplied by 1 upon 2 to the power minus 2, multiplied by 3 to the power minus 1, multiplied by 1 upon 6 to the power minus 1. We need to express this as a rational number. So first of all look at this expression 2 upon 3 whole q and this whole square for this we will use this law. So using this law we would get 2 upon 3 whole to the power 3 multiplied by 2 that is 6 and this divided by 2 upon 3 whole square multiplied by 1 upon 2 to the power minus 2 multiplied by 3 upon 6 whole to the power minus 1. Now here you can see that 3 2 times is 6 so we get 2 upon 3 whole to the power 6 divided by 2 upon 3 whole to the power 2 multiplied by 1 upon 2 whole to the power minus 2 multiplied by 1 upon 2 whole to the power minus 1. Now to divide 2 upon 3 whole to the power 6 by 2 upon 3 whole to the power 2 we will use this law. Now as in this case 6 is greater than 2 that is m is greater than n so this expression would be equal to a upon v to the power m minus n. So here we would get this is equal to 2 upon 3 whole to the power 6 minus 2 and this is multiplied by 1 upon 2 whole to the power minus 2 minus 1. For this multiplication of these two rational numbers we have used this law that is a upon v to the power m multiplied by a upon v to the power n is equal to a upon v to the power m plus n. So here we have added the parts of the rational number 1 upon 2 so this is equal to 2 upon 3 whole to the power 4 multiplied by 1 upon 2 whole to the power minus 3. This further gives us 2 upon 3 whole to the power 4 multiplied by 2 to the power 3 as we know that a upon v to the power minus n is equal to b upon a to the power n. So this is equal to 2 to the power 4 multiplied by 2 to the power 3 upon 3 to the power 4 or you can say this is equal to 2 to the power 7 upon 3 to the power 4 which is equal to 128 upon 81. So this is our final answer 128 upon 81 the given expression is expressed as rational number 128 upon 81. So this completes the session hope you have understood the solution for this question.