Express result in specific form

A question is this type if and only if it asks to use binomial expansion to express a result in a specific form like a + b√2 or with integer coefficients.

5 questions · Moderate -0.4

1.04a Binomial expansion: (a+b)^n for positive integer n
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OCR C2 Q6
9 marks Moderate -0.5
  1. (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. \(( 1 - \sqrt { 2 } ) ^ { 8 }\)
    3. The second and fifth terms of an arithmetic sequence are 26 and 41 repectively.
AQA C2 2005 June Q6
9 marks Moderate -0.8
6
  1. 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.
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Expand \((1 + x)^4\) in ascending powers of \(x\). [2]
  2. 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. \((1 - \sqrt{2})^8\) [7]
AQA AS Paper 1 2020 June Q4
3 marks Moderate -0.5
In the binomial expansion of \((\sqrt{3} + \sqrt{2})^4\) there are two irrational terms. Find the difference between these two terms. [3 marks]
WJEC Unit 1 Specimen Q10
8 marks Standard +0.8
  1. Use the binomial theorem to express \(\left(\sqrt{3} - \sqrt{2}\right)^5\) in the form \(a\sqrt{3} + b\sqrt{2}\), where \(a\), \(b\) are integers whose values are to be found. [5]
  2. Given that \(\left(\sqrt{3} - \sqrt{2}\right)^5 \approx 0\), 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. [3]