 Hello students, let's work out the following problem. It says using differentials find the approximate value of the following and the given number is 32.15 to the power 1 by 5. So let's now start the solution and first of all we define or we assume y as a function of x and we define it to be x to the power 1 by 5 and we take x in such a way so that we can easily find out its fifth root and that is near to 32.15 so we take x to be 32 and since we need to have x plus delta x is equal to 32.15 we take delta x to be 0.15 so x plus delta x is equal to 32.15. Now we know that y is equal to f of x plus delta x minus fx now f of x plus delta x will be 32.15 to the power 1 by 5 minus f of x f of x would be 32 to the power 1 by 5 and this is the value of delta y not y. Now we have to find the value of 32.15 to the power 1 by 5 so from this we have 32.15 to the power 1 by 5 is equal to delta y plus 32 to the power 1 by 5 now 32 can be written as 2 to the power 5 and it's power 1 by 5, 5 gets cancelled with 5 and we have 32.15 to the power 1 by 5 equal to delta y plus 2. Now we have to find the value of delta y now we know that delta y is approximately equal to dy and dy is equal to dy by dx into delta x so delta y will be equal to dy by dx into delta x now y is x to the power 1 by 5 so delta y would be dy by dx would be 1 by 5 into x to the power 1 by 5 minus 1 that is minus 4 by 5 into delta x delta x is 0.15 now this is 1 by 5 into x to the power minus 1 by 5 now x is 32 it's power minus 4 by 5 into 0.15 again this is 1 by 5 into 2 to the power 5 to the power minus 4 by 5 into 0.15 this is again equal to 1 by 5 into 2 to the power minus 4 into 0.15 now 0.15 divided by 5 gives us 0.03 and this is equal to 0.03 upon 2 to the power 4 2 to the power 4 is 16 and dividing 0.03 by 16 gives us 0.00187 now 32.15 to the power 1 by 5 is equal to delta y plus 2 now delta y is this so it is 0.00187 plus 2 that is equal to 2.00187 hence the approximate value of 32.15 to the power 1 by 5 is 2.00187 so this completes the question and the session by for now take care have a good day