 Hello friends, let's work out the following problem. It says using differentials find approximate value of the following up to three places of decimal. The given number is 3.968 to the power 3 by 2. Let's now proceed on to the solution and let's define y as a function of x. Let's define y equal to x to the power 3 by 2 and here we take x in such a way so that we can easily find out the square root. And since we need to have x plus delta x is equal to 3.968 we take delta x as minus of 0.032 so x plus delta x is equal to 3.968. Now we know that delta y is equal to f of x plus delta x minus fx so this is x plus delta x to the power 3 by 2 minus x to the power 3 by 2. Now x plus delta x is 3.968 to the power 3 by 2 minus x to the power 3 by 2. Now this is 3.968 to the power 3 by 2 minus 4 can be written as 2 square. 2 gets cancelled with 2 and we have 3.968 to the power 3 by 2 minus 8 so this implies 3.968 to the power 3 by 2 equal to delta y plus 8. 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 is equal to dy by dx into delta x. Now dy by dx will be 3 by 2 into x to the power 3 by 2 minus 1 that is 1 by 2 into delta x and delta x is minus 0.032. Let's now substitute the value of x so we have 3 by 2 into 4 to the power 1 by 2 into minus 0.032. Now square root of 4 is 2 so this is 3 by 2 into 2 into minus 0.032. Now minus 0.03 into 3 is equal to minus 0.096. Now 3.968 to the power 3 by 2 is equal to delta y plus 8 now delta y is minus 0.096 plus 8. So this is equal to 7.904 hence the answer is 7.904. So this completes the question and the session by for now take care have a good day.