Express result in specific form

\begin{enumerate} \setcounter{enumi}{5} \item (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
(b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.

1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\) \item The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.

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1. Using the binomial expansion, or otherwise, express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
2. 1. Hence show that \(( 1 + \sqrt { 5 } ) ^ { 4 } = 56 + 24 \sqrt { 5 }\).
   2. Hence show that \(\log _ { 2 } ( 1 + \sqrt { 5 } ) ^ { 4 } = k + \log _ { 2 } ( 7 + 3 \sqrt { 5 } )\), where \(k\) is an integer.

4. (a) Expand \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
(b) Using your expansion, express each of the following in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are integers.

1. \(( 1 + \sqrt { 2 } ) ^ { 4 }\)
2. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)

10. (a) Use the binomial theorem to express \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 }\) in the form \(a \sqrt { 3 } + b \sqrt { 2 }\), where \(a , b\) are integers whose values are to be found.
(b) Given that \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 5 } \approx 0\), use your answer to part (a) to find an approximate value for \(\sqrt { 6 }\) in the form \(\frac { c } { d }\), where \(c\) and \(d\) are positive integers whose values are to be found.

4 In the binomial expansion of \(( \sqrt { } 3 + \sqrt { } 2 ) ^ { 4 }\) there are two irrational terms. Find the difference between these two terms.