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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 \(5^2 - \sqrt{49}\).

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    Model answer

    \(5^2 = 25\) and \(\sqrt{49} = 7\), so \(25 - 7 = 18\).

    Mark scheme

    • \(25\) or \(7\) seen — M1
    • 18 — A1
  2. 2 Write down [1 mark]

    Write down the value of \(2^5\).

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    Model answer

    \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

    Mark scheme

    • 32 — B1
  3. 3 Simplify [2 marks]

    Write each of these as a single power. (a) \(5^3 \times 5^4\) [1 mark] (b) \((6^2)^5\) [1 mark]

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    Model answer

    (a) Add the indices: \(5^{3+4} = 5^7\). (b) Multiply the indices: \(6^{2 \times 5} = 6^{10}\).

    Mark scheme

    • (a) \(5^7\) — B1
    • (b) \(6^{10}\) — B1
  4. 4 Work out [2 marks]

    Work out the value of \(5^0 + 2^{-2}\). Give your answer as a mixed number.

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    Model answer

    \(5^0 = 1\) and \(2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}\). So \(1 + \dfrac{1}{4} = 1\dfrac{1}{4}\).

    Mark scheme

    • \(5^0 = 1\) or \(2^{-2} = \dfrac{1}{4}\) — M1
    • \(1\dfrac{1}{4}\) or \(\dfrac{5}{4}\) or \(1.25\) — A1
  5. 5 Show that [2 marks]

    Show that \(2^6 \times 4^2 = 2^{10}\).

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    Model answer

    \(4 = 2^2\), so \(4^2 = (2^2)^2 = 2^4\). Then \(2^6 \times 2^4 = 2^{6+4} = 2^{10}\).

    Mark scheme

    • \(4^2 = 2^4\) or \(4 = 2^2\) — M1
    • \(2^6 \times 2^4 = 2^{10}\) with a clear conclusion — Q1
  6. 6 Work out [4 marks]

    Work out the value of each of these. (a) \(16^{\frac{3}{4}}\) [2 marks] (b) \(\left(\dfrac{1}{9}\right)^{-\frac{1}{2}}\) [2 marks]

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    Model answer

    (a) \(16^{\frac{1}{4}} = 2\), then \(2^3 = 8\). (b) The negative index flips the fraction: \(\left(\dfrac{1}{9}\right)^{-\frac{1}{2}} = 9^{\frac{1}{2}} = 3\).

    Mark scheme

    • (a) \(\sqrt[4]{16} = 2\) or \(16^{\frac{1}{4}} = 2\) — M1
    • (a) 8 — A1
    • (b) \(9^{\frac{1}{2}}\) — M1
    • (b) 3 — A1
  7. 7 Simplify [3 marks]

    Simplify \(\dfrac{(2x^3)^2}{4x^2}\).

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    Model answer

    \((2x^3)^2 = 4x^6\). Then \(\dfrac{4x^6}{4x^2} = x^4\).

    Mark scheme

    • \((2x^3)^2 = 4x^6\) — M1
    • Divides the numbers and the powers of \(x\) — M1
    • \(x^4\) — 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 \(2 \times 2 \times 2 = 8\).