Solution to Exercise 8.4-1

The problem was to approximate 258^1/4.

Step 1: 258 is very close to 256, which is 4⁴. You might have had some trouble with this one, but since you should know that you are looking for perfect fourth powers (that is squares of squares), you should be able to find this quickly by a simple exhaustive search:

   (12)2  =   12  =    1

   (22)2  =   42  =   16

   (32)2  =   92  =   81

   (42)2  =  162  =  256

Step 2: Find the derivative of the fourth root function.

   f(x)  =  x1/4

             1
   f'(x)  =    x-3/4
             4

Step 3: Find the value of the derivative at the x where you know the fourth root. This would be at  x = 256.

             1             1    1      1
   f'(4)  =    256-3/4  =    ×     =     
             4             4   43     256

Step 4: What is the difference between the number you know the fourth root of and the number you want to take the fourth root of? This will be your h.

   h  =  258 - 256  =  2

Step 5: Apply the approximation formula.

                                 1
   2581/4  »  4 + hf'(256)  =  4     =  4.0078125
                                128

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