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      Problem solving with the laws of indices

      Lesson details

      Learning outcome

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

      Key learning points

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

      Keywords

      • Power - An alternative word for exponent or index. Used when reading exponents above 3. E.g. 2^4 is read as “2 to the power of 4”.

      • Vinculum - Vinculum is the horizontal line placed over an expression to show that everything below that line is one group.

      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.

      Teacher tip

      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.

      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

      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

      6 Questions

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

      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

      To help you plan your 10 maths lesson on: Problem solving with the laws of indices, download all teaching resources for free and adapt to suit your pupils' needs...