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Grade 10 · Core Mathematics · Lesson 3 of 13

Discovering Product and Quotient Laws

A Grade 10 Core Mathematics CBE lesson from the sub-strand Indices and Logarithms. In it, learners use factor strings to discover product and quotient laws.

Strand Numbers and AlgebraSub-strand Indices and LogarithmsLesson code 10CM-12-L340 minutes · 3 slides

KICD learning outcome (b)

Derive the laws of indices using factors.

By the end of this lesson, learners should be able to

  1. Use factor strings to discover product and quotient laws.
  2. Explain why exponents add or subtract for common bases.
  3. Apply each law to a new numerical expression.

Key vocabulary

Product law
A same-base multiplication rule that adds exponents.
For base seven, 7² × 7³ equals 7⁵.
Common base
The identical factor appearing in two powers.
Both 4³ and 4² have four as their common base.
Quotient law
A same-base division rule that subtracts exponents.
8⁵ divided by 8² simplifies to 8³.
Cancel
Remove equal nonzero factors from numerator and denominator.
Two matching factors cancel before the remaining power is written.
Exponent law
A rule connecting operations on powers with exponent arithmetic.
The quotient law changes 9⁷ ÷ 9³ into 9⁴.
Verify
Check an algebraic result using another valid method.
Expand two short powers to verify the product rule.

Lesson preview

Slide 1 · Multiplication joins factor stringsOpening · 12 min
A diagram showing how multiplying two copies of three by four copies of three makes a combined string of six threes.
Figure 1: A diagram showing how multiplying two copies of three by four copies of three makes a combined string of six threes.

When powers share a base, multiplying them joins their repeated-factor strings. Counting every copy shows why exponents add; the rule is not a shortcut to memorise without understanding its factor structure. Use Figure 1 to observe how repeated-factor strings join together when multiplying powers.

  • Equal bases allow factor strings to be joined directly.
  • The first exponent counts factors in the first string.
  • The second exponent counts factors in the second string.
  • Joining strings explains addition of exponents precisely.
Worked example. Confirm both powers have the common base 3. Add the exponents: 2 + 4 = 6. Write 3^2 × 3^4 = 3^6. Answer: 3^6.

Check question: Which statement correctly uses factors to explain why 52×53=55?

  1. Joining a string of two 5s and a string of three 5s gives a combined string of five 5s.
  2. Multiplying the exponents gives 6, so the result should actually be 56.
  3. Adding the two bases gives 10, which then halves to give a base of 5.
  4. Multiplying the two bases gives 25, so the result should actually be 255.
Show answer

Joining a string of two 5s and a string of three 5s gives a combined string of five 5s. Expanding both powers gives (5×5)×(5×5×5). Counting the total number of 5s gives five, which is written as 55. The base remains unchanged while the counts of repeated factors add.

Slide 2 · Division removes matching factorsModel · 14 min
A visual model showing how cancelling two sevens from both the numerator and denominator leaves three sevens, giving seven cubed.
Figure 2: A visual model showing how cancelling two sevens from both the numerator and denominator leaves three sevens, giving seven cubed.

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

  • All 3 slides with diagrams and class activities
  • Teacher notes, SNE adaptations and assessment checks
  • Homework and the other 12 lessons in this sub-strand
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All 13 lessons in Indices and Logarithms

  1. 1Reading Repeated Factors as IndicesExpress numbers in index form
  2. 2Index Forms with a Specified BaseExpress numbers in index form
  3. 3Discovering Product and Quotient LawsDerive the laws of indices using factors
  4. 4Power of a Power and Identity CasesDerive the laws of indices using factors
  5. 5Indices in Measurement and DesignApply the laws of indices in different situations
  6. 6From Powers of Ten to Common LogsRelate index notation to logarithm notation to base 10
  7. 7Interpreting Base-Ten LogarithmsRelate index notation to logarithm notation to base 10
  8. 8Reading Common Logarithms in TablesDetermine common logarithms of numbers from mathematical tables and calculators
  9. 9Comparing Table and Calculator LogarithmsDetermine common logarithms of numbers from mathematical tables and calculators
  10. 10Logarithms for Multiplication and DivisionApply common logarithms in multiplication, division, powers and roots of numbers
  11. 11Logarithms for Powers and RootsApply common logarithms in multiplication, division, powers and roots of numbers
  12. 12Integrated Logarithmic ComputationApply common logarithms in multiplication, division, powers and roots of numbers
  13. 13Why Indices and Logarithms MatterAppreciate the use of indices and common logarithms in mathematical computations