 Hello and welcome to the session. Let's work out the following question. It says using differential find the approximate value of the following the given number is 33 to the power minus 1 by 5 So let's now move on to the solution and let us first express Y as a function of X So we take Y as X to the power 1 by 5 and we choose X in such a way so that we can easily find out its fifth root and Which is near to 33. So we take X to be 32 So delta X would be 1 because we want that X plus delta X is 33. So X plus delta X will be equal to 33 Now delta Y is equal to F of X plus delta X minus F X now F X is X to the power 1 by 5. So F of X plus delta X would be 32 plus 1 whole to the power 1 by 5 minus F X that is 30 to the power 1 by 5 Since here we have 33 to the power minus 1 by 5. Here also we need to choose Y equal to X to the power minus 1 by 5 and here also it becomes 33 to the power minus 1 by 5 and here it becomes 30 to the power minus 1 by 5 So this is equal to 33 to the power minus 1 by 5 minus 32 to the power minus 1 by 5. So this is delta Y Now we have to find the value of 33 to the power minus 1 by 5. So this implies 33 to the power minus 1 by 5 is equal to 33 to the power minus 1 by 5 will be equal to delta Y plus 32 to the power minus 1 by 5 now 32 can be written as 2 to the power 5 and It's power minus 1 by 5 This is again equal to delta Y plus 1 upon 2 to the power 5 to the power 1 by 5 5 gets cancelled with 5 and we are left with delta Y plus 1 by 2 now delta Y is approximately equal to dy dy is equal to dy by dx into delta X Now Y is X to the power minus 1 by 5 So this is equal to dy by dx is equal to minus 1 by 5 into X to the power minus 1 by 5 minus 1 into delta X or delta X is 1 So this is again equal to minus 1 by 5 into X to the power minus 6 by 5 into 1 This is equal to minus 1 by 5 into X to the power 6 by 5 Now X is 32 which can be written as 2 to the power 5 to the power 6 by 5 5 gets cancelled with 5 here and we are left with minus 1 by 5 into 2 to the power 6 This is equal to minus 1 by 5 into 2 to the power 6 5 into 2 to the power 6 is 320 now 33 to the power minus 1 by 5 is equal to delta Y Which is equal to minus 1 by 320 plus 1 by 2 Now taking LCM and simplifying LCM would be 320 and the numerator we have minus 1 plus 160 that is equal to 159 upon 320 and that is further equal to 0.497 Hence the approximate value of 33 to the power minus 1 by 5 is equal to 0.497 So this completes the question and the session. Bye for now. Take care. Have a good day