Square Roots w/ Binomial Theorem??

The binomial series is a generalization of the binomial theorem.

See this link: http://en.wikipedia.org/wiki/Binomial_series

Case 1: Let alpha = 1/2. The series converges for |x| < 1. Let y = 1 + x or x = y - 1. The series converges for 0 < y < 2. So if you want to calculate the sqrt of y in this range:

y^(1/2) = (1 + (y - 1))^(1/2) = (1 + x)^(1/2)

Then you expand the LHS with the Binomial series to get the answer for 0 < y < 2.

Case 2: Let alpha = -1/2. The series converges for |x| < 1. Let y = 1 / (1 + x) or x = (1/y) - 1. The series converges for 1/2 < y < infinity. To calculate y in this range:

y^(1/2) = (1 + x)^(-1/2)

Then you expand the LHS by the binomial series to get the answer for 1/2 < y < infinity.

The sum of the squares of the two factors is comparable to the sq. of the hypotenuse. in simple terms remember what the scarecrow mentioned on the Wizard of oz., appropriate after he gained "brains".