Rationalize denominator simple

A question is this type if and only if it asks to rationalize a denominator of the form a/(b + c√d) or a/(b - c√d) by multiplying by the conjugate, resulting in the form p + q√r.

79 questions · Easy -1.1

1.02b Surds: manipulation and rationalising denominators
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AQA AS Paper 2 2022 June Q7
4 marks Moderate -0.8
7 The expression $$\frac { 3 - \sqrt { } n } { 2 + \sqrt { } n }$$ can be written in the form \(a + b \sqrt { } n\), where \(a\) and \(b\) and \(n\) are rational but \(\sqrt { } n\) is irrational. Find expressions for \(a\) and \(b\) in terms of \(n\).
AQA Paper 1 2023 June Q7
4 marks Moderate -0.3
7
  1. Given that \(n\) is a positive integer, express $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ as a single fraction not involving surds.
    7
  2. Hence, deduce that $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ is a rational number for all positive integer values of \(n\)
Edexcel PURE 2024 October Q2
Easy -1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
  2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.
WJEC Unit 1 2018 June Q1
Moderate -0.8
Showing all your working, simplify a) \(\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }\),
b) \(\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }\).
Edexcel C1 Q5
6 marks Easy -1.2
  1. Write \(\sqrt{45}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \(\frac{2(3 + \sqrt{5})}{(3 - \sqrt{5})}\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [5]
Edexcel C1 Q6
4 marks Easy -1.3
  1. Expand and simplify \((4 + \sqrt{3})(4 - \sqrt{3})\). [2]
  2. Express \(\frac{26}{4 + \sqrt{3}}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
Edexcel C1 Q1
5 marks Easy -1.2
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q40
6 marks Moderate -0.8
Giving your answers in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational numbers, find
  1. \((3 - \sqrt{8})^2\), [3]
  2. \(\frac{1}{4 - \sqrt{8}}\). [3]
Edexcel M2 2014 January Q1
4 marks Easy -1.2
Simplify fully
  1. \((2\sqrt{x})^2\) [1]
  2. \(\frac{5 + \sqrt{7}}{2 + \sqrt{7}}\) [3]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
Edexcel C1 Q2
5 marks Moderate -0.8
Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,
  1. find the value of \(a\) and the value of \(b\). [2]
Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,
  1. find the value of \(c\) and the value of \(d\). [3]
OCR C1 2006 June Q2
6 marks Easy -1.2
  1. Evaluate \(27^{\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1 - \sqrt{5}}{3 + \sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]
OCR MEI C1 Q9
5 marks Moderate -0.8
Simplify \((3 + \sqrt{2})(3 - \sqrt{2})\). Express \(\frac{1 + \sqrt{2}}{3 - \sqrt{2}}\) in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are rational. [5]
OCR MEI C1 2006 January Q8
5 marks Easy -1.3
  1. Simplify \(5\sqrt{8} + 4\sqrt{50}\). Express your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
  2. Express \(\frac{\sqrt{3}}{6 - \sqrt{3}}\) in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are rational. [3]
OCR MEI C1 2012 June Q5
5 marks Moderate -0.8
  1. Simplify \(\frac{10\sqrt{6}}{3}{\sqrt{24}}\). [3]
  2. Simplify \(\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}\). [2]
Edexcel C1 Q2
3 marks Moderate -0.8
Express $$\frac{2}{3\sqrt{5} + 7}$$ in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are rational. [3]
Edexcel C1 Q9
10 marks Moderate -0.3
  1. Express each of the following in the form \(p + q\sqrt{2}\) where \(p\) and \(q\) are rational.
    1. \((4 - 3\sqrt{2})^2\)
    2. \(\frac{1}{2 + \sqrt{2}}\) [5]
    1. Solve the equation $$y^2 + 8 = 9y.$$
    2. Hence solve the equation $$x^3 + 8 = 9x^{\frac{1}{2}}.$$ [5]
OCR MEI C1 Q4
5 marks Easy -1.2
  1. Write \(\sqrt{48} + \sqrt{3}\) in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
  2. Simplify \(\frac{1}{5 + \sqrt{2}} + \frac{1}{5 - \sqrt{2}}\). [3]
OCR MEI C1 Q7
5 marks Moderate -0.8
  1. Simplify \(\frac{10(\sqrt{6})^3}{\sqrt{24}}\). [3]
  2. Simplify \(\frac{1}{4 - \sqrt{5}} + \frac{1}{4 + \sqrt{5}}\). [2]
OCR MEI C1 Q4
5 marks Easy -1.2
  1. Find the value of \(144^{-\frac{1}{2}}\). [2]
  2. Simplify \(\frac{1}{5 + \sqrt{7}} + \frac{4}{5 - \sqrt{7}}\). Give your answer in the form \(\frac{a + b\sqrt{7}}{c}\). [3]
OCR PURE Q1
5 marks Easy -1.3
In this question you must show detailed reasoning.
  1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
  2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
WJEC Unit 1 2019 June Q07
6 marks Moderate -0.8
Given that \(a\), \(b\) are integers, simplify the following. Show all your working.
  1. \(\frac{2\sqrt{3} + a}{\sqrt{3} - 1}\) [3]
  2. \(\frac{2\sqrt{6b^2} - \sqrt{27} + \sqrt{192}}{\sqrt{2}}\) [3]
WJEC Unit 1 2022 June Q2
6 marks Moderate -0.3
Showing all your working, simplify the following expression. [6] $$5\sqrt{48} + \frac{2+5\sqrt{3}}{5+3\sqrt{3}} - (2\sqrt{3})^3$$
SPS SPS SM 2020 October Q4
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
  2. Solve the equation \((8p^6)^{\frac{1}{3}} = 8\). [3]
SPS SPS SM 2022 February Q1
6 marks Easy -1.3
  1. Evaluate \(27^{-\frac{2}{3}}\). [2]
  2. Express \(5\sqrt{5}\) in the form \(5^n\). [1]
  3. Express \(\frac{1-\sqrt{5}}{3+\sqrt{5}}\) in the form \(a + b\sqrt{5}\). [3]