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      The laws of indices - multiplication

      Lesson details

      Learning outcome

      I can use the laws of indices to multiply two powers where the bases are the same.

      Key learning points

      1. When multiplying two terms, you can sometimes write this more simply.
      2. If the powers have the same base, then the powers can be combined into a single power.
      3. The exponent or index of the new power reflects this combination.
      4. By studying the structure of multiplication, you can see how the index will change.
      5. a^b × a^c = a^(b+c)

      Keywords

      • Power - 16 is the fourth power of 2. Alternatively this can be written as 2^4 which is read as “2 to the power of 4”.

      • Index - An exponent is a number positioned above and to the right of a base value. It indicates repeated multiplication. An alternative word for this is index (plural indices).

      • Coefficient - A numerical coefficient is a constant multiplier of the variables in a term.

      • Commutative - An operation is commutative if the values it is operating on can be written in either order without changing the calculation.

      • Associative - An operation is associative if a repeated application of the operation produces the same result regardless of how pairs of values are grouped.

      Common misconception

      When multiplying terms with coefficients, pupils also add the coefficients as well as the exponents.

      Pupils should be encouraged to rewrite their expression using the associative and commutative laws, with the number parts grouped and powers grouped. This hopefully avoids this error as they can see it is the product of the numbers.

      Teacher tip

      When changing the base, quick recall of powers of 2, 3, 4, 5 and 10 are useful. Using MWB you call out a number and the pupils write it as a power. This could also be done as a bingo game.

      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.
      Calculate $$-3 \times 5^2$$

      Correct Answer: -75

      Q2.
      Calculate $$-3 \times 4^2$$

      Correct Answer: -48

      Q3.
      Calculate $$-3 \times 2^2$$

      Correct Answer: -12

      Q4.
      Calculate $$-4 \times 4^2$$

      Correct Answer: -64

      Q5.
      Calculate $$3^2 + (-4)^2$$

      Correct Answer: 25

      Q6.
      Calculate $$9^2 + (-5)^2$$

      Correct Answer: 106

      6 Questions

      Q1.
      What is the value of $$a$$ for $$3^3 \times 3^a = 3^{20}$$?

      Correct Answer: 17

      Q2.
      What is the value of $$a$$ for $$3^3 \times 3^a = 3^7$$?

      Correct Answer: 4

      Q3.
      What is the value of $$a$$ for $$5^8 \times 5^a = 5^{10}$$?

      Correct Answer: 2

      Q4.
      What is the value of $$a$$ for $$3^3 \times a^7 = 3^{10}$$?

      Correct Answer: 3

      Q5.
      What is the value of $$a$$ for $$3^7 \times a^8 = 3^{15}$$?

      Correct Answer: 3

      Q6.
      What is the value of $$a$$ for $$6^3 \times 6^8 = 6^a$$?

      Correct Answer: 11

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