Expand and simplify surd expressions

A question is this type if and only if it asks to expand brackets containing surds and simplify, such as (3 + √5)(3 - √5) or (5 - √8)(1 + √2), typically resulting in the form a + b√c.

31 questions · Easy -1.2

1.02b Surds: manipulation and rationalising denominators
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Edexcel C12 Specimen Q3
6 marks Easy -1.3
3. Answer this question without the use of a calculator and show all your working.
  1. Show that $$( 5 - \sqrt { 8 } ) ( 1 + \sqrt { 2 } ) \equiv a + b \sqrt { 2 }$$ giving the values of the integers \(a\) and \(b\).
  2. Show that $$\sqrt { 80 } + \frac { 30 } { \sqrt { 5 } } \equiv c \sqrt { 5 } , \text { where } c \text { is an integer. }$$
Edexcel C1 2007 January Q2
4 marks Easy -1.3
2. (a) Express \(\sqrt { } 108\) in the form \(a \sqrt { } 3\), where \(a\) is an integer.
(b) Express \(( 2 - \sqrt { 3 } ) ^ { 2 }\) in the form \(b + c \sqrt { 3 }\), where \(b\) and \(c\) are integers to be found.
Edexcel C1 2007 June Q1
2 marks Easy -1.8
Simplify \(( 3 + \sqrt { } 5 ) ( 3 - \sqrt { } 5 )\). \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-02_108_93_2614_1786}
Edexcel C1 2009 January Q3
2 marks Easy -1.8
Expand and simplify \(( \sqrt { } 7 + 2 ) ( \sqrt { } 7 - 2 )\).
Edexcel C1 2013 January Q3
6 marks Easy -1.3
  1. Express $$( 5 - \sqrt { } 8 ) ( 1 + \sqrt { } 2 )$$ in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
  2. Express $$\sqrt { } 80 + \frac { 30 } { \sqrt { } 5 }$$ in the form \(c \sqrt { } 5\), where \(c\) is an integer.
OCR MEI C1 2015 June Q6
5 marks Easy -1.2
6
  1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
  2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 Q3
4 marks Easy -1.2
3 Write \(( \sqrt { 3 } - \sqrt { 2 } ) ^ { 2 }\) in the form \(a + b \sqrt { 6 }\) where \(a\) and \(b\) are integers to be determined.
OCR MEI C1 Q7
5 marks Moderate -0.8
7
  1. Express \(( 2 + \sqrt { 3 } ) ^ { 2 }\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers to be determined.
  2. Given that \(x\) and \(y\) are integers, prove that \(\frac { 1 } { x - \sqrt { y } } + \frac { 1 } { x + \sqrt { y } }\) can be written in the form \(\frac { p } { q }\) where \(p\) and \(q\) are both integers.
OCR C1 Q3
4 marks Easy -1.2
3. (i) Express \(\frac { 18 } { \sqrt { 3 } }\) in the form \(k \sqrt { 3 }\).
(ii) Express \(( 1 - \sqrt { 3 } ) ( 4 - 2 \sqrt { 3 } )\) in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are integers.
OCR C1 Q4
5 marks Moderate -0.8
4. 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. \(\frac { 1 } { 2 + \sqrt { 2 } }\)
OCR MEI C1 Q1
5 marks Easy -1.2
1
  1. Expand and simplify \(( 3 + 4 \sqrt { 5 } ) ( 3 - 2 \sqrt { 5 } )\).
  2. Express \(\sqrt { 72 } + \frac { 32 } { \sqrt { 2 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 Q2
5 marks Moderate -0.8
2
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
OCR MEI C1 Q7
5 marks Easy -1.2
7
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 Q10
5 marks Easy -1.2
10
  1. Simplify \(\frac { \sqrt { 48 } } { 2 \sqrt { 27 } }\).
  2. Expand and simplify \(( 5 - 3 \sqrt { 2 } ) ^ { 2 }\).
OCR MEI C1 2012 January Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 + 3 \sqrt { 2 } ) ( 5 - 2 \sqrt { 2 } )\).
  2. Simplify \(\sqrt { 54 } + \frac { 12 } { \sqrt { 6 } }\).
OCR MEI C1 2014 June Q4
5 marks Easy -1.2
4
  1. Expand and simplify \(( 7 - 2 \sqrt { 3 } ) ^ { 2 }\).
  2. Express \(\frac { 20 \sqrt { 6 } } { \sqrt { 50 } }\) in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.
AQA C1 2005 January Q5
7 marks Easy -1.2
5
  1. Simplify \(( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )\).
  2. Express \(\sqrt { 12 }\) in the form \(m \sqrt { 3 }\), where \(m\) is an integer.
  3. Express \(\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
AQA C1 2012 January Q3
9 marks Easy -1.2
3
    1. Simplify \(( 3 \sqrt { 2 } ) ^ { 2 }\).
    2. Show that \(( 3 \sqrt { 2 } - 1 ) ^ { 2 } + ( 3 + \sqrt { 2 } ) ^ { 2 }\) is an integer and find its value.
  2. Express \(\frac { 4 \sqrt { 5 } - 7 \sqrt { 2 } } { 2 \sqrt { 5 } + \sqrt { 2 } }\) in the form \(m - \sqrt { n }\), where \(m\) and \(n\) are integers.
AQA C1 2005 June Q5
5 marks Easy -1.2
5 Express each of the following in the form \(m + n \sqrt { 3 }\), where \(m\) and \(n\) are integers:
  1. \(( \sqrt { 3 } + 1 ) ^ { 2 }\);
  2. \(\frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 }\).
AQA C1 2006 June Q4
6 marks Easy -1.2
4
  1. Express \(( 4 \sqrt { 5 } - 1 ) ( \sqrt { 5 } + 3 )\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
  2. Show that \(\frac { \sqrt { 75 } - \sqrt { 27 } } { \sqrt { 3 } }\) is an integer and find its value.
AQA C1 2010 June Q2
6 marks Moderate -0.8
2
  1. Express \(( 3 - \sqrt { 5 } ) ^ { 2 }\) in the form \(m + n \sqrt { 5 }\), where \(m\) and \(n\) are integers.
  2. Hence express \(\frac { ( 3 - \sqrt { 5 } ) ^ { 2 } } { 1 + \sqrt { 5 } }\) in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers.
    (4 marks)
Edexcel C2 Q2
7 marks Moderate -0.3
2. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.
Edexcel C1 Q2
4 marks Easy -1.3
  1. Express \(\sqrt{108}\) in the form \(a\sqrt{3}\), where \(a\) is an integer. [1]
  2. Express \((2 - \sqrt{3})^2\) in the form \(b + c\sqrt{3}\), where \(b\) and \(c\) are integers to be found. [3]
Edexcel C1 Q1
2 marks Easy -1.8
Simplify \((3 + \sqrt{5})(3 - \sqrt{5})\). [2]
Edexcel C1 Specimen Q3
4 marks Easy -1.3
  1. Express \(\sqrt{80}\) in the form \(a\sqrt{5}\), where \(a\) is an integer. [1]
  2. Express \((4 - \sqrt{5})^2\) in the form \(b + c\sqrt{5}\), where \(b\) and \(c\) are integers. [3]