A Level Maths Notes: C2 – The Binomial Theorem 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 find
to
5 decimal places. We can substitute
and
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: Find
to
10 decimal places. We can substitute
and
into
the expression for the binomial expansion.
Further expansion is pointless since we only need 10 decimal places.
We have
