Choosing a suitable value for binomial approximation

Example 1

If the expansion is of other forms, can try the method shown in example 2.

Example 2

Suppose you are asked to expand  using binomial expansion, then use the results to approximate 

How do you select a suitable value for the approximation?

Step 1

Make sure the value selected is within the range of values for which the expansion is valid. In this example, the valid range is from -4 to 4.

Step 2

4-x = (13 / perfect square) or (perfect square/13)

So 4-x = (13 /4 or 13/9) or (49/13)

x= 3/13 or 3/4 or 23/9

Choose the smallest value 3/13 as value closer to 0 gives better approximation.

Therefore, 

LHS can be approximated using the binomial expansion. Hence  can be approximated.