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Indices

discia.org 10 questions · 31 marks
Name Date
  1. Non-calculator 3 marks
    (a)

    Simplify \(t^{ 7 } \times t^{ 6 }\)

    (b)

    Simplify \(t^{ 12 } \div t^{ 4 }\)

    (c)

    Simplify \((t^{ 2 })^{ 2 }\)

    Show answer
    (a) \(t^{ 13 }\)
    (b) \(t^{ 8 }\)
    (c) \(t^{ 4 }\)
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  2. Non-calculator 3 marks
    (a)

    Write down the value of \(5^{0}\)

    (b)

    Work out \(3^{ -2 }\)

    Give your answer as a fraction.

    (c)

    Write \(\frac{1}{ k^{ 3 } }\) as a power of \(k\)

    Show answer
    (a) \(1\)
    (b) \(\frac{1}{ 9 }\)
    (c) \(k^{ -3 }\)
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  3. Non-calculator 4 marks
    (a)

    Simplify fully \(4w^{ 2 } \times 2w^{ 3 }\)

    (b)

    Simplify fully \(\frac{ 8w^{ 5 } }{ 4w^{ 2 } }\)

    Show answer
    (a) \(8w^{ 5 }\)
    (b) \(2w^{ 3 }\)
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  4. Non-calculator 2 marks
    (a)

    \(10^{ x } = \frac{1}{1000}\)

    Calculate the value of \(x\).

    (b)

    \(3^{ x } = 27\)

    Calculate the value of \(x\).

    Show answer
    (a) \(-3\)
    (b) \(3\)
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  5. Non-calculator 3 marks

    Hannah and Oscar are each checking some index working.

    (a)

    Hannah writes \((7^{ 8 })^{ 6 } = 7^{ 14 }\)

    Is Hannah correct? Give a reason for your answer.

    (b) Oscar wants to simplify \(7t^{ 4 } \times 5t^{ 2 }\) He says, “Add 7 and 5, then multiply 4 and 2.” His answer is \(12t^{ 8 }\) Write correct or not correct for each part of his method, and show the working that decides each one.
    Part of the methodCorrect or not correct
    Add 7 and 5
    Multiply 4 and 2
    Show answer
    (a) No: a power of a power multiplies the indices, \(8 \times 6 = 48\), so \((7^{ 8 })^{ 6 } = 7^{ 48 }\)
    (b) Add 7 and 5: not correct. Multiply 4 and 2: not correct.
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  6. Non-calculator 2 marks

    Simplify fully \(\frac{ (p^{ 2 })^{ 4 } }{ p^{ 6 } }\)

    Show answer
    \(p^{ 2 }\)
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  7. Non-calculator 3 marks

    Work out the value of \((2 \times 2^{ 3 }) \div (2^{ 2 } \times 2^{ 4 })\)

    Give your answer as a decimal.

    Show answer
    \(0.25\)
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  8. Non-calculator 5 marks

    Give each answer as a single power of the prime number named in the part.

    (a)

    Write \(16 \times 2^{ 7 } \times 2^{ 3 }\) as a single power of 2.

    (b)

    Simplify \(32^{ 3 } \times 16^{ 4 }\)

    Give your answer in the form \(2^{ m }\), where \(m\) is an integer.

    Show answer
    (a) \(2^{ 14 }\)
    (b) \(2^{ 31 }\)
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  9. Non-calculator 2 marks

    \(16t^{ 6 } = ( at^{ 3 } )^{ b }\)

    where \(a\) and \(b\) are positive integers.

    Work out the value of \(a\) and the value of \(b\).

    Show answer
    \(a = 4\) and \(b = 2\)
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  10. Non-calculator 4 marks
    (a)

    Simplify fully \(6c^{ 4 }d^{ 3 } \times 3c^{ 3 }d^{ 3 }\)

    (b)

    Simplify fully \(\frac{ 36c^{ 8 }d^{ 7 } }{ 6c^{ 4 }d^{ 3 } }\)

    Show answer
    (a) \(18c^{ 7 }d^{ 6 }\)
    (b) \(6c^{ 4 }d^{ 4 }\)
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discia.org Indices · sheet 1
Answers Indices · sheet 1
  1. (a) \(t^{ 13 }\)
    (b) \(t^{ 8 }\)
    (c) \(t^{ 4 }\)
  2. (a) \(1\)
    (b) \(\frac{1}{ 9 }\)
    (c) \(k^{ -3 }\)
  3. (a) \(8w^{ 5 }\)
    (b) \(2w^{ 3 }\)
  4. (a) \(-3\)
    (b) \(3\)
  5. (a) No: a power of a power multiplies the indices, \(8 \times 6 = 48\), so \((7^{ 8 })^{ 6 } = 7^{ 48 }\)
    (b) Add 7 and 5: not correct. Multiply 4 and 2: not correct.
  6. \(p^{ 2 }\)
  7. \(0.25\)
  8. (a) \(2^{ 14 }\)
    (b) \(2^{ 31 }\)
  9. \(a = 4\) and \(b = 2\)
  10. (a) \(18c^{ 7 }d^{ 6 }\)
    (b) \(6c^{ 4 }d^{ 4 }\)
discia.org Indices · sheet 1