Question 6 - A-Level Maths

Exam Board	AQA
Module	C2 (Core Mathematics 2)
Year	2005
Session	June
Topic	Binomial Theorem (positive integer n)
Type	Express result in specific form

Using the binomial expansion, or otherwise, express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
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.