13. Compute the square root of 40.

The closest square to 40 is 36. We will use the fact that

and then we can use the binomial theorem.

We identify the numbers to substitute in for the variables in this formula.

a = 36, b = 4, n = 1/2, r = 0, 1, 2, 3, 4, ...

Since the square root of 36 is 6, and negative exponents give us reciprocals, this simplifies to

When we convert to decimals we get the following arithmetic

The square root of 40 is closer to 6.32455532... Note that we are a little high with the estimate we see above. However, the next term will be negative and the result after the next term will be a little low. By taking more and more terms from the binomial theorem we can get closer and closer to the real answer.