Rationalize denominator two surds

A question is this type if and only if it asks to rationalize a denominator containing two different surds, such as (8 - √15)/(2√3 + √5), resulting in the form a√m + b√n.

3 questions · Moderate -0.8

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Edexcel P1 2023 October Q3
6 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Write \(\frac { 8 - \sqrt { 15 } } { 2 \sqrt { 3 } + \sqrt { 5 } }\) in the form \(a \sqrt { 3 } + b \sqrt { 5 }\) where \(a\) and \(b\) are integers to be found.
  2. Hence, or otherwise, solve $$( x + 5 \sqrt { 3 } ) \sqrt { 5 } = 40 - 2 x \sqrt { 3 }$$ giving your answer in simplest form.
OCR MEI C1 2010 June Q5
5 marks Easy -1.3
  1. Express \(\sqrt{48} + \sqrt{27}\) in the form \(a\sqrt{3}\). [2]
  2. Simplify \(\frac{5\sqrt{7}}{3 - \sqrt{2}}\). Give your answer in the form \(\frac{b + c\sqrt{7}}{d}\). [3]
AQA AS Paper 1 2024 June Q3
4 marks Moderate -0.8
Express \(\frac{\sqrt{3} + 3\sqrt{5}}{\sqrt{5} - \sqrt{3}}\) in the form \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers. Fully justify your answer. [4 marks]