## The Binomial Expansion and Estimation

The binomial expansion can in certain circumstances give highly accurate estimates of certain powers or roots. This can happen for example when we want to find not too high a power of a number that is close to an integer.

The binomial expansion is given by

Suppose we are asked to findto 5 decimal places. We can substituteand into the expression for the binomial expansion.

Further expansion is pointless since we only need 5 decimal places. After simplifying we have

to 5 decimal places.

Example: Findto 10 decimal places. We can substituteand into the expression for the binomial expansion.

Further expansion is pointless since we only need 10 decimal places.

We have