[discia] GCSE Maths › Number › Surds

Surds

discia.org 10 questions · 41 marks
Name Date
  1. Non-calculator 4 marks
    (a)

    Write \(\sqrt{ 54 }\) in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

    (b)

    Simplify \(4\sqrt{ 32 }\)

    Give your answer in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

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    (a) \(3\sqrt{ 6 }\)
    (b) \(16\sqrt{ 2 }\)
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  2. Non-calculator 2 marks

    \(\sqrt{n} = 7\sqrt{ 5 }\)

    Calculate the value of \(n\).

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    \(245\)
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  3. Non-calculator 3 marks
    \(a = 3\) and \(b = 12\) Match each expression to its value. One has been done for you.
    ExpressionValue
    \(\sqrt{a} + \sqrt{b}\)\(3\sqrt{3}\)
    \(\sqrt{b}\) 
    \(a\sqrt{b}\) 
    \(\sqrt{ab}\) 
    Values
    \(2\sqrt{3}\)\(4\)\(3\sqrt{3}\)\(6\)\(6\sqrt{3}\)\(12\sqrt{3}\)\(36\)
    Show answer
    \(\sqrt{b}\) matches \(2\sqrt{3}\), \(a\sqrt{b}\) matches \(6\sqrt{3}\), \(\sqrt{ab}\) matches \(6\)
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  4. Non-calculator 3 marks

    The diagram shows a right-angled triangle. All dimensions are in centimetres.

    x√82√7

    Find the value of \(x\).

    Give your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

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    \(5\sqrt{ 3 }\ \text{cm}\)
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  5. Non-calculator 6 marks
    (a)

    Simplify fully \(\sqrt{ 275 } + \sqrt{ 44 }\)

    Give your answer in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

    (b)

    Simplify fully \(\sqrt{ 75 } - \sqrt{ 12 }\)

    Give your answer in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

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    (a) \(7\sqrt{ 11 }\)
    (b) \(3\sqrt{ 3 }\)
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  6. Non-calculator 7 marks
    (a)

    Expand and simplify \(\sqrt{ 3 }\left(\sqrt{ 3 } + 5\right)\)

    (b)

    Expand and simplify \(\left(2 + \sqrt{ 10 }\right)\left(3 - \sqrt{ 10 }\right)\)

    (c)

    Expand and simplify \(\left(2 + \sqrt{ 5 }\right)\left(2 - \sqrt{ 5 }\right)\)

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    (a) \(3 + 5\sqrt{ 3 }\)
    (b) \(-4 +\sqrt{ 10 }\)
    (c) \(-1\)
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  7. Non-calculator 4 marks
    (a)

    Rationalise the denominator of \(\frac{ 28 }{\sqrt{ 7 } }\)

    Give your answer in its simplest form.

    (b)

    Rationalise the denominator and simplify \(\frac{ 2 }{\sqrt{ 7 } }\)

    Show answer
    (a) \(4\sqrt{ 7 }\)
    (b) \(\frac{ 2\sqrt{ 7 } }{ 7 }\)
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  8. Non-calculator 3 marks

    Show that \(\frac{ 7\sqrt{ 7 } }{\sqrt{ 5 } } - \frac{\sqrt{ 20 } }{\sqrt{ 7 } }\) can be written in the form \(\frac{ m\sqrt{ n } }{ k }\), where \(m\), \(n\) and \(k\) are integers.

    Show answer
    The terms simplify to \(\frac{ 7\sqrt{ 35 } }{ 5 }\) and \(\frac{ 2\sqrt{ 35 } }{ 7 }\). Subtracting over a denominator of \(35\) gives \(\frac{ 39\sqrt{ 35 } }{ 35 }\), so \(m = 39\), \(n = 35\) and \(k = 35\).
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  9. Non-calculator 3 marks

    Show that \(\frac{ \sqrt{ 27 } - \sqrt{ 3 } }{ 2\sqrt{ 3 } }\) simplifies to an integer.

    Show answer
    \(\sqrt{ 27 } = 3\sqrt{ 3 }\), so the top is \(2\sqrt{ 3 }\) and \(\frac{ 2\sqrt{ 3 } }{ 2\sqrt{ 3 } } = 1\), an integer.
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  10. Non-calculator 6 marks
    (a)

    Rationalise the denominator of \(\frac{ 2 }{ 3 + \sqrt{ 7 } }\)

    Give your answer in the form \(p - q\sqrt{ 7 }\), where \(p\) and \(q\) are integers.

    (b)

    Rationalise the denominator of \(\frac{ 46 }{ 6 - \sqrt{ 13 } }\)

    Give your answer in the form \(p + q\sqrt{ 13 }\), where \(p\) and \(q\) are integers.

    Show answer
    (a) \(3 - \sqrt{ 7 }\)
    (b) \(12 + 2\sqrt{ 13 }\)
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discia.org Surds · sheet 1
Answers Surds · sheet 1
  1. (a) \(3\sqrt{ 6 }\)
    (b) \(16\sqrt{ 2 }\)
  2. \(245\)
  3. \(\sqrt{b}\) matches \(2\sqrt{3}\), \(a\sqrt{b}\) matches \(6\sqrt{3}\), \(\sqrt{ab}\) matches \(6\)
  4. \(5\sqrt{ 3 }\ \text{cm}\)
  5. (a) \(7\sqrt{ 11 }\)
    (b) \(3\sqrt{ 3 }\)
  6. (a) \(3 + 5\sqrt{ 3 }\)
    (b) \(-4 +\sqrt{ 10 }\)
    (c) \(-1\)
  7. (a) \(4\sqrt{ 7 }\)
    (b) \(\frac{ 2\sqrt{ 7 } }{ 7 }\)
  8. The terms simplify to \(\frac{ 7\sqrt{ 35 } }{ 5 }\) and \(\frac{ 2\sqrt{ 35 } }{ 7 }\). Subtracting over a denominator of \(35\) gives \(\frac{ 39\sqrt{ 35 } }{ 35 }\), so \(m = 39\), \(n = 35\) and \(k = 35\).
  9. \(\sqrt{ 27 } = 3\sqrt{ 3 }\), so the top is \(2\sqrt{ 3 }\) and \(\frac{ 2\sqrt{ 3 } }{ 2\sqrt{ 3 } } = 1\), an integer.
  10. (a) \(3 - \sqrt{ 7 }\)
    (b) \(12 + 2\sqrt{ 13 }\)
discia.org Surds · sheet 1