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Exam questions · Maths · Number Without a Calculator

Powers, Roots and Indices

  • 7 exam questions
  • 16 marks
  • 10 quick checks
  1. 1 Work out [2 marks]

    Work out the value of (a) \(4^3\) (1 mark) (b) \(\sqrt{121}\) (1 mark)

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    Model answer

    (a) \(4 \times 4 \times 4 = 64\). (b) \(\sqrt{121} = 11\) because \(11 \times 11 = 121\).

    Mark scheme

    • (a) 64 — B1
    • (b) 11 — B1
  2. 2 Simplify [2 marks]

    Simplify (a) \(7^4 \times 7^5\) (1 mark) (b) \(3^{10} \div 3^4\) (1 mark)

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    Model answer

    (a) Add the indices: \(7^{4+5} = 7^9\). (b) Subtract the indices: \(3^{10-4} = 3^6\).

    Mark scheme

    • (a) \(7^9\) — B1
    • (b) \(3^6\) — B1
  3. 3 Work out [3 marks]

    (a) Write down the value of \(9^0\). (1 mark) (b) Work out the value of \(5^{-2}\). (2 marks)

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    Model answer

    (a) Any non-zero number to the power 0 is 1, so \(9^0 = 1\). (b) \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).

    Mark scheme

    • (a) 1 — B1
    • (b) \(\dfrac{1}{5^2}\) or \(\dfrac{1}{25}\) — M1
    • (b) \(\dfrac{1}{25}\) or 0.04 — A1
  4. 4 Work out [2 marks]

    Work out the value of \(\dfrac{3^5 \times 3^{-2}}{3}\).

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    Model answer

    The top is \(3^{5-2} = 3^3\). Then \(3^3 \div 3^1 = 3^2 = 9\).

    Mark scheme

    • \(3^3\) for the numerator, or a power of 3 with at most one index error — M1
    • 9 — A1
  5. 5 Simplify [2 marks]

    Simplify fully \((2p^3q)^4\).

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    Model answer

    Raise every part to the power 4: \(2^4 \times (p^3)^4 \times q^4 = 16p^{12}q^4\).

    Mark scheme

    • Any two of \(16\), \(p^{12}\), \(q^4\) correct — M1
    • \(16p^{12}q^4\) — A1
  6. 6 Work out [3 marks]

    (a) Work out the value of \(49^{\frac{1}{2}}\). (1 mark) (b) Work out the value of \(8^{-\frac{2}{3}}\). (2 marks)

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    Model answer

    (a) A power of one half means the square root, so \(49^{\frac{1}{2}} = 7\). (b) \(8^{-\frac{2}{3}} = \dfrac{1}{8^{\frac{2}{3}}} = \dfrac{1}{(\sqrt[3]{8})^2} = \dfrac{1}{2^2} = \dfrac{1}{4}\).

    Mark scheme

    • (a) 7 — B1
    • (b) \(\dfrac{1}{8^{2/3}}\) or \(\sqrt[3]{8} = 2\) — M1
    • (b) \(\dfrac{1}{4}\) or 0.25 — A1
  7. 7 Write [2 marks]

    Write \(\dfrac{1}{32}\) as a power of 2.

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    Model answer

    \(32 = 2^5\), so \(\dfrac{1}{32} = \dfrac{1}{2^5} = 2^{-5}\).

    Mark scheme

    • \(32 = 2^5\) or \(\dfrac{1}{2^5}\) — M1
    • \(2^{-5}\) — A1

Quick check

  1. 1

    Between which two whole numbers does \(\sqrt{40}\) lie?

    1. A6 and 7
    2. B5 and 6
    3. C4 and 5
    4. D7 and 8
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    A: 6 and 7

    \(6^2 = 36\) and \(7^2 = 49\), and 40 is between 36 and 49.

  2. 2

    Write \(16 \times 8\) as a single power of 2.

    1. A\(4^7\)
    2. B\(2^{24}\)
    3. C\(2^{12}\)
    4. D\(2^7\)
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    D: \(2^7\)

    \(16 = 2^4\) and \(8 = 2^3\), so \(2^4 \times 2^3 = 2^7\).

  3. 3

    What is the value of \(\sqrt{196}\)?

    1. A98
    2. B14
    3. C16
    4. D13
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    B: 14

    14 × 14 = 196.

  4. 4

    What is the value of \(4^3\)?

    1. A12
    2. B81
    3. C64
    4. D256
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    C: 64

    \(4 \times 4 \times 4 = 64\), not \(4 \times 3 = 12\).

  5. 5

    Simplify \(a^5 \times a^3\).

    1. A\(a^2\)
    2. B\(a^{15}\)
    3. C\(a^8\)
    4. D\(2a^8\)
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    C: \(a^8\)

    Add the indices when multiplying: \(a^{5+3} = a^8\).

  6. 6

    Simplify \((3^2)^4\).

    1. A\(3^8\)
    2. B\(9^8\)
    3. C\(3^6\)
    4. D\(3^{16}\)
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    A: \(3^8\)

    For a power of a power, multiply the indices: \(3^{2 \times 4} = 3^8\).

  7. 7

    What is the value of \(5^{-2}\)?

    1. A\(-10\)
    2. B\(\dfrac{1}{25}\)
    3. C\(-25\)
    4. D\(\dfrac{1}{10}\)
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    B: \(\dfrac{1}{25}\)

    A negative index means a reciprocal: \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).

  8. 8

    What is the value of \(7^0\)?

    1. A1
    2. BUndefined
    3. C0
    4. D7
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    A: 1

    Any non-zero number to the power 0 is 1.

  9. 9

    Simplify \(2^6 \div 2^{-2}\).

    1. A\(2^4\)
    2. B\(2^{-3}\)
    3. C\(2^8\)
    4. D\(2^{-12}\)
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    C: \(2^8\)

    Subtract the indices: \(6 - (-2) = 8\), so the answer is \(2^8\).

  10. 10

    What is the value of \(8^{\frac{1}{3}}\)?

    1. A24
    2. B\(\dfrac{8}{3}\)
    3. C\(\dfrac{1}{8}\)
    4. D2
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    D: 2

    A power of one third means the cube root, and \(2 \times 2 \times 2 = 8\).