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