5. Compute

We can rewrite this as

We choose 25 because it is the closest square to 26. Now we can use the binomial theorem. The first term is just the first term inside the parentheses raised to the power.

and it begins to become clear why we wanted to have the first term be a perfect square. The second term in the binomial theorem is has the power as a coefficient, the first term raised to one smaller power and the second term raised to the first power. One smaller than 1/2 is -1/2, and we get

In the next term the coefficient has an second consecutive factor in the top and a 2! in the bottom. The first term is raised to one less power and the second term is raised to one more power.

We see some patterns beginning to appear. The next coefficient on top is the exponent on the 25 in the term before, and the power on the second term gives us the factorial in the bottom.The next term is

Let us start to work on this before we go any further. Recall that negative exponents give us recoprocals.

Notice also that since each subsequent term after the second one has another negative factor in its coefficient, the signs will alternate between positive and negative. Exponents with halves in the denominator give us square roots.

If we change these fractions to decimals and do the arithmetic we get.

Our approximation is now well within 4 places after the decimal of the square root of 26 which is approximately