[discia] GCSE Maths › Number › Powers and Roots

Powers and Roots

discia.org 10 questions · 22 marks
Name Date
  1. Either paper 2 marks
    (a)

    Find the value of \(10^2\)

    (b)

    Find the value of \(9^3\)

    Show answer
    (a) \(100\)
    (b) \(729\)
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  2. Either paper 3 marks
    (a)

    Write \(3 \times 3 \times 3 \times 3\) in index form.

    (b)

    Find the value of \(2^3\)

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    (a) \(3^4\)
    (b) \(8\)
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  3. Either paper 1 mark

    The square root of \(x\) is \(3\).

    Work out the value of \(x\).

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    \(9\)
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  4. Either paper 1 mark

    Circle the value of \(4^{ 3 } + 7^{ 2 }\)

    \(113\)     \(121\)     \(599\)     \(3136\)

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    \(113\)
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  5. Either paper 3 marks
    (a)

    Work out the value of \(\sqrt{ 196 }\)

    (b)

    Work out the value of \(\sqrt[3]{ 216 }\)

    (c)

    Work out the value of \(\sqrt{ 196 } \times \sqrt[3]{ 216 }\)

    Show answer
    (a) \(14\)
    (b) \(6\)
    (c) \(84\)
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  6. Either paper 2 marks

    Show that 661 is the sum of a power of 5 and a square number.

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    \(661 = 625 + 36 = 5^{ 4 } + 6^2\)
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  7. Either paper 3 marks

    Show that exactly sixty-eight of the 4-digit numbers are square numbers.

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    \(31^2 = 961\) is too small and \(100^2 = 10000\) is too big, so the bases run from \(32\) to \(99\): \(68\) numbers
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  8. Either paper 3 marks

    The diagrams show the first two cube numbers, 1 and 8.

    1 cm1st cube number: 1 2 cm2nd cube number: 8

    Show that the sixth cube number is the sum of three different primes.

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    e.g. \(2 + 17 + 197 = 216\), which is \(6^3\), and \(2\), \(17\) and \(197\) are all prime and all different (any other valid triple is also correct)
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  9. Either paper 2 marks

    Work out the value of \(\left(2\frac{1}{4}\right)^2\)

    Give your answer as a mixed number.

    Show answer
    \(5\frac{1}{16}\)
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  10. Non-calculator 2 marks

    Do not use a calculator.

    (a) Here are four numbers.
    999991,3318,100
    Circle the number that is a square number.
    (b) Here are four more numbers.
    974147162
    Circle the number that is half of a square number.
    Show answer
    (a) \(8100\)
    (b) \(162\)
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discia.org Powers and Roots · sheet 1
Answers Powers and Roots · sheet 1
  1. (a) \(100\)
    (b) \(729\)
  2. (a) \(3^4\)
    (b) \(8\)
  3. \(9\)
  4. \(113\)
  5. (a) \(14\)
    (b) \(6\)
    (c) \(84\)
  6. \(661 = 625 + 36 = 5^{ 4 } + 6^2\)
  7. \(31^2 = 961\) is too small and \(100^2 = 10000\) is too big, so the bases run from \(32\) to \(99\): \(68\) numbers
  8. e.g. \(2 + 17 + 197 = 216\), which is \(6^3\), and \(2\), \(17\) and \(197\) are all prime and all different (any other valid triple is also correct)
  9. \(5\frac{1}{16}\)
  10. (a) \(8100\)
    (b) \(162\)
discia.org Powers and Roots · sheet 1