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Lesson 13 of 10

    Advanced problem solving with the laws of indices

    I can use my knowledge of the laws of indices to solve problems.

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    Lesson 13 of 10
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      Advanced problem solving with the laws of indices

      I can use my knowledge of the laws of indices to solve problems.

      Lesson details

      Key learning points

      1. Surds can be written in index notation.
      2. The base could be a number, letter or combination of the two.
      3. The laws are another way you can simplify an expression.
      4. Plotting numbers against their squared values can help you estimate unknown amounts.

      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

      Using the laws of indices, when dividing by a number with a negative exponent pupils can incorrectly subtract the negative exponent.

      When the bases of all the numbers are the same, forming an equation with the exponents can reduce errors.

      Using MWB, give the students an answer e.g. 5^6, and they need to form a complicated question using as many laws of indices as they can so the answer simplifies to 5^6. For more advanced question creations, pupils can used fractional and/or negative indices within the laws.
      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).

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      Prior knowledge starter quiz

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      6 Questions

      Q1.
      Evaluate $$16^\frac{1}{2}$$, giving the positive solution where necessary.

      Correct Answer: 4

      Q2.
      Evaluate $$9^\frac{3}{2}$$, giving the positive solution where necessary.

      Correct Answer: 27

      Q3.
      Evaluate $$25^\frac{1}{2}$$, giving the positive solution where necessary.

      Correct Answer: 5

      Q4.
      What is the value of $$b$$ for $$b^7 \div b^5 =9^2$$?

      Correct Answer: 9

      Q5.
      What is the value of $$k$$ for $$0.2^5 \div 0.2^k = 0.2^7$$?

      Correct Answer: -2

      Q6.
      What number is written as $$2^3 \times 5 \times 7$$ as a product of its prime factors?

      Correct Answer: 280

      Assessment exit quiz

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      6 Questions

      Q1.
      What is the value of $$x$$ for $$(3^3)^4 × 3^2 = 3^x$$ ?

      Correct Answer: 14

      Q2.
      What is the value of $$x$$ for $$\frac{2^{12} \times 2^4}{2^{10}} = 2^x $$?

      Correct Answer: 6

      Q3.
      What is the value of $$x$$ for $$\frac{5^{14} \times 5^5}{5^{-2}} = 5^x $$?

      Correct Answer: 21

      Q4.
      What is the value of $$x$$ for $$3^2 \times 3^6 = 9^x$$ ?

      Correct Answer: 4

      Q5.
      What is the value of $$x$$ for $$2^3 \times 4^3 = 2^x$$ ?

      Correct Answer: 9

      Q6.
      What is the value of $$x^{4x}$$ for $$2^x \times 2^{3x} = x^{4x}$$ ?

      Correct Answer: 256

      Lesson appears in

      UnitMaths / Arithmetic procedures: index laws

      Maths

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