Show surd expression equals value

A question is this type if and only if it asks to prove or show that a specific surd expression simplifies to a given value or form, such as showing (√180 - √80)/√5 is an integer.

17 questions · Easy -1.1

1.02b Surds: manipulation and rationalising denominators
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Edexcel P1 2022 June Q3
5 marks Moderate -0.8
3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
  1. Show that \(\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }\) is an integer and find its value.
  2. Simplify $$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$ giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational numbers.
Edexcel C12 2014 January Q4
7 marks Moderate -0.8
4. Answer this question without the use of a calculator and show all your working.
  1. Show that $$\frac { 4 } { 2 \sqrt { 2 } - \sqrt { 6 } } = 2 \sqrt { 2 } ( 2 + \sqrt { 3 } )$$
  2. Show that $$\sqrt { 27 } + \sqrt { 21 } \times \sqrt { 7 } - \frac { 6 } { \sqrt { 3 } } = 8 \sqrt { 3 }$$
Edexcel C12 2016 January Q2
5 marks Easy -1.2
2. (i) Given that \(\frac { 49 } { \sqrt { 7 } } = 7 ^ { a }\), find the value of \(a\).
(ii) Show that \(\frac { 10 } { \sqrt { 18 } - 4 } = 15 \sqrt { 2 } + 20\) You must show all stages of your working.
Edexcel C12 2016 October Q3
5 marks Easy -1.2
3. Answer this question without the use of a calculator and show your method clearly.
  1. Show that $$\sqrt { 45 } - \frac { 20 } { \sqrt { 5 } } + \sqrt { 6 } \sqrt { 30 } = 5 \sqrt { 5 }$$
  2. Show that $$\frac { 17 \sqrt { 2 } } { \sqrt { 2 } + 6 } = 3 \sqrt { 2 } - 1$$
Edexcel C12 2018 October Q1
5 marks Easy -1.2
  1. (i) Given that \(125 \sqrt { 5 } = 5 ^ { a }\), find the value of \(a\).
    (ii) Show that \(\frac { 16 } { 4 - \sqrt { 8 } } = 8 + 4 \sqrt { 2 }\)
You must show all stages of your working.
Edexcel C1 2012 June Q3
5 marks Moderate -0.8
3. Show that \(\frac { 2 } { \sqrt { } ( 12 ) - \sqrt { } ( 8 ) }\) can be written in the form \(\sqrt { } a + \sqrt { } b\), where \(a\) and \(b\) are integers.
OCR MEI AS Paper 2 2019 June Q3
3 marks Moderate -0.8
3 Without using a calculator, prove that \(3 \sqrt { 2 } > 2 \sqrt { 3 }\).
OCR MEI AS Paper 2 2023 June Q5
3 marks Easy -1.8
5 Show that the distance between the points \(( 5,2 )\) and \(( 11 , - 1 )\) is \(a \sqrt { b }\), where \(a\) and \(b\) are integers to be determined.
OCR MEI Paper 2 2018 June Q1
2 marks Easy -1.8
1 Show that \(\sqrt { 27 } + \sqrt { 192 } = a \sqrt { b }\), where \(a\) and \(b\) are prime numbers to be determined.
AQA C1 2010 January Q4
7 marks Easy -1.2
4
  1. Show that \(\frac { \sqrt { 50 } + \sqrt { 18 } } { \sqrt { 8 } }\) is an integer and find its value.
    (3 marks)
  2. Express \(\frac { 2 \sqrt { 7 } - 1 } { 2 \sqrt { 7 } + 5 }\) in the form \(m + n \sqrt { 7 }\), where \(m\) and \(n\) are integers.
    (4 marks)
Edexcel C1 Q3
4 marks Easy -1.2
  1. Find the integer \(n\) such that
$$4 \sqrt { 12 } - \sqrt { 75 } = \sqrt { n }$$
Pre-U Pre-U 9794/2 2015 June Q1
3 marks Easy -1.8
1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).
OCR MEI C1 Q1
4 marks Moderate -0.5
You are given that \(a = \frac{3}{2}\), \(b = \frac{9 - \sqrt{17}}{4}\) and \(c = \frac{9 + \sqrt{17}}{4}\). Show that \(a + b + c = abc\). [4]
AQA AS Paper 1 2019 June Q4
4 marks Moderate -0.8
Show that \(\frac{\sqrt{6}}{\sqrt{3} - \sqrt{2}}\) can be expressed in the form \(m\sqrt{n} + n\sqrt{m}\), where \(m\) and \(n\) are integers. Fully justify your answer. [4 marks]
AQA AS Paper 1 Specimen Q4
3 marks Easy -1.2
Show that \(\frac{5\sqrt{2} + 2}{3\sqrt{2} + 4}\) can be expressed in the form \(m + n\sqrt{2}\), where \(m\) and \(n\) are integers. [3 marks]
AQA Paper 1 2024 June Q7
4 marks Standard +0.3
Show that $$\frac{3 + \sqrt{8n}}{1 + \sqrt{2n}}$$ can be written as $$\frac{4n - 3 + \sqrt{2n}}{2n - 1}$$ where \(n\) is a positive integer. [4 marks]
Pre-U Pre-U 9794/2 2011 June Q2
6 marks Easy -1.2
  1. Expand and simplify \((7 - 2\sqrt{3})^2\). [2]
  2. Show that $$\frac{\sqrt{125}}{2 + \sqrt{5}} = 25 - 10\sqrt{5}.$$ [4]