- (a) Find the binomial expansion of
$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } } , \quad | x | < \frac { 4 } { 5 }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
Give each coefficient in its simplest form.
(b) Find the exact value of \(( 4 + 5 x ) ^ { \frac { 1 } { 2 } }\) when \(x = \frac { 1 } { 10 }\)
Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined.
(c) Substitute \(x = \frac { 1 } { 10 }\) into your binomial expansion from part (a) and hence find an approximate value for \(\sqrt { 2 }\)
Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.