Factoring out constants first

2. (a) Find the binomial expansion of \(( 1 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\).
(2 marks)
(b) (i) Find the binomial expansion of \(( 27 + 6 x ) ^ { - \frac { 1 } { 3 } }\) up to and including the term in \(x ^ { 2 }\), simplifying the coefficients.
(ii) Given that \(\sqrt [ 3 ] { \frac { 2 } { 7 } } = \frac { 2 } { \sqrt [ 3 ] { 28 } }\), use your binomial expansion from part (b)(i) to obtain an approximation to \(\sqrt [ 3 ] { \frac { 2 } { 7 } }\), giving your answer to six decimal places.
(2 marks)

5. $$f ( x ) = \frac { 10 } { \sqrt { 4 - 3 x } }$$

1. Show that the first 4 terms in the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), are $$A + B x + C x ^ { 2 } + \frac { 675 } { 1024 } x ^ { 3 }$$ where \(A , B\) and \(C\) are constants to be found. Give each constant in simplest form. Given that this expansion is valid for \(| x | < k\)
2. state the largest value of \(k\). By substituting \(x = \frac { 1 } { 3 }\) into \(\mathrm { f } ( x )\) and into the answer for part (a),
3. find an approximation for \(\sqrt { 3 }\) Give your answer in the form \(\frac { a } { b }\) where \(a\) and \(b\) are integers to be found.
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1. Use the binomial theorem to expand

$$\sqrt { } ( 4 - 9 x ) , \quad | x | < \frac { 4 } { 9 } ,$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), simplifying each term.