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Lesson 12 of 10

    The laws of indices - fractional exponents

    I can use the laws of indices with fractional exponents.

    Lesson 12 of 10
    New
    New

      The laws of indices - fractional exponents

      I can use the laws of indices with fractional exponents.

      Lesson details

      Key learning points

      1. All of the index laws can be applied with fractional exponents.
      2. Using a calculator can help you to form an idea about how to evaluate a power containing a fractional exponent.
      3. You can reason that a power with a fractional exponent is equivalent to finding the square root of the base.
      4. √a = a^(1/2)

      Keywords

      • Reciprocal - A reciprocal is the multiplicative inverse of any non-zero number. Any non-zero number multiplied by its reciprocal is equal to 1.

      Common misconception

      Pupils multiply the base by the fractional exponent. e.g 25^(1/2) = 12.5

      The use of a calculator allows pupils to see the index of a fraction is not to be multiplied by the base. Embedding the laws of indices where two bases are the same and each number has an index of 1/2 helps recognize the 1/2 index as a square root.

      Using MWB, put the number 100 in the centre. Pupils must create as many numbers with an exponent that equate to 10. E.g 10^2, (1/100)^-1, 1000^(2/3), 4x125^(2/3), etc. Display the summary of the laws of indices to helps and give access to calculators for more support.
      Teacher tip

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

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      6 Questions

      Q1.
      What is the value of $$x$$ for $$(m^x)^8 = m^{32}$$

      Correct Answer: 4

      Q2.
      What is the value of $$x$$ for $$(m^{-2})^5 = m^x$$

      Correct Answer: -10

      Q3.
      What is the value of $$x$$ for $$(m^8)^x = m^{80}$$

      Correct Answer: 10

      Q4.
      What is the value of $$x$$ for $$\frac{1}{16} = 2^x$$

      Correct Answer: -4

      Q5.
      What is the value of $$x$$ for $$\frac{1}{64} = x^{-6}$$

      Correct Answer: 2

      Q6.
      What is the value of $$a$$ for $$7^9 \times 7^a = 7^6$$?

      Correct Answer: -3

      Assessment exit quiz

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      6 Questions

      Q1.
      Evaluate $$36^{\frac{1}{2}}$$, giving the positive solution where necessary.

      Correct Answer: 6

      Q2.
      Evaluate $$1000^{\frac{1}{3}}$$, giving the positive solution where necessary.

      Correct Answer: 10

      Q3.
      Evaluate $$27^{\frac{2}{3}}$$, giving the positive solution where necessary.

      Correct Answer: 9

      Q4.
      Evaluate $$1000^{\frac{2}{3}}$$, giving the positive solution where necessary.

      Correct Answer: 100

      Q5.
      Evaluate $$16^{\frac{3}{4}}$$, giving the positive solution where necessary.

      Correct Answer: 8

      Q6.
      Evaluate $$16^{\frac{3}{2}}$$, giving the positive solution where necessary.

      Correct Answer: 64

      Lesson appears in

      UnitMaths / Arithmetic procedures: index laws