Surds worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

1. 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.

2. 2 marks

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

   Calculate the value of \(n\).

3. 6 marks
   (a)

   Write \(\sqrt{ 27 } + \sqrt{ 12 }\) in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

   (b)

   Write \(\sqrt{ 396 } - \sqrt{ 99 }\) in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

4. 7 marks
   (a)

   Multiply out the brackets and simplify \(\sqrt{ 13 }\left(\sqrt{ 13 } + 5\right)\)

   (b)

   Multiply out the brackets and simplify \(\left(3 + \sqrt{ 7 }\right)\left(2 - \sqrt{ 7 }\right)\)

   (c)

   Multiply out the brackets and simplify \(\left(6 + \sqrt{ 6 }\right)\left(6 - \sqrt{ 6 }\right)\)

5. 4 marks
   (a)

   Rationalise the denominator of \(\frac{ 20 }{\sqrt{ 5 } }\)

   Give your answer in its simplest form.

   (b)

   Rationalise the denominator of \(\frac{ 9 }{\sqrt{ 5 } }\)

6. 6 marks
   (a)

   Rationalise the denominator and simplify \(\frac{ 39 }{ 4 + \sqrt{ 3 } }\)

   Give your answer in the form \(p - q\sqrt{ 3 }\), where \(p\) and \(q\) are integers.

   (b)

   Rationalise the denominator and simplify \(\frac{ 34 }{ 6 - \sqrt{ 2 } }\)

   Give your answer in the form \(p + q\sqrt{ 2 }\), where \(p\) and \(q\) are integers.

7. 5 marks
   (a)

   Write \(\sqrt{ 20 } + \sqrt{ 5 }\) as a single surd \(\sqrt{c}\), where \(c\) is an integer.

   (b)

   Explain why \(\sqrt{ 3 } + \sqrt{ 5 }\) cannot be written as a single surd in the same way.

8. 6 marks

   A rectangle has width \(\sqrt{ 90 }\) cm and length \(\sqrt{ 160 }\) cm.

   (a)

   Calculate the area of the rectangle.

   (b)

   Calculate the perimeter of the rectangle.

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

9. 4 marks
   (a)

   Simplify \(\sqrt{ 147 }\) fully.

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

   (b)

   Write \(4\sqrt{ 275 }\) in the form \(a\sqrt{b}\), where \(b\) is as small as possible.

10. 2 marks

    \(\sqrt{n} = 3\sqrt{ 7 }\)

    Find the value of \(n\).

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