Grade 10 essential mathematics Numbers and Algebra – Indices Notes

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Essential Mathematics — 1.0 Numbers & Algebra
Subtopic 1.2: Indices (for learners aged ~15, Kenya)
Specific Learning Outcomes
  1. Identify and outline the sub-sub-strands:
    • Indices (powers and bases) — what a^n means: base a and index (exponent) n.
    • Laws of indices — rules for multiplying, dividing and powering expressions with indices.
    • Applications of indices — uses in area/volume, scientific notation, growth, computing, etc.
  2. Express numbers in index form in different situations (squares, cubes, repeated multiplication, scientific notation).
  3. Deduce the four main laws of indices from the meaning of powers (using expanded multiplication/division).
  4. Apply the laws of indices in numerical computations and simple algebraic calculations.
  5. Recognise where indices are used in practical computations (e.g., area = side2, volume = side3, scientific notation).
Key Definitions

Power / index / exponent: In an, a is the base and n is the index (exponent). It means "multiply a by itself n times": an = a × a × ... × a (n factors).

Examples: 23 = 2×2×2 = 8 ; 52 = 5×5 = 25 ; 71 = 7.

Special cases: a0 = 1 (for a ≠ 0); a-n = 1 / an.

Expressing numbers in index form
  • Repeated multiplication: 3×3×3×3 = 34.
  • Squares and cubes: area of square with side 6 cm is 62 cm2. Volume of cube side 2 m is 23 m3.
  • Scientific notation (useful for very large/small numbers):
    6 000 000 = 6 × 106; 0.00045 = 4.5 × 10-4.
  • Converting products: 4×4×4×4×4 = 45. Converting repeated division: 8 ÷ 8 ÷ 8 = 8-2 (check: 8-2 = 1/82 = 1/64).
Deduce the four main laws of indices (with reasoning)
Law 1 — Product rule
Start with am × an. Expand:
am = a × a × ... (m times), an = a × a × ... (n times).
When multiplied you have m + n factors of a, so:
am × an = am+n
Example: 23 × 24 = 27 = 128.
Law 2 — Quotient rule
Start with am ÷ an where m ≥ n. Expand and cancel common factors:
am ÷ an = am-n
(If m < n, result is a negative index: am-n = 1 / an-m.) Example: 57 ÷ 53 = 54 = 625.
Law 3 — Power of a power
(am)n means multiply am by itself n times: am × am × ... (n times) = am·n.
(am)n = amn
Example: (23)2 = 23×2 = 26 = 64.
Law 4 — Power of a product
(ab)n = an bn. This follows because (ab)(ab)...(ab) = a×a×... × b×b×... with n copies of each. Example: (3×4)2 = 122 = 144 and 32×42 = 9×16 = 144.
Notes: From the quotient rule and power-of-product you can also obtain the rules for negative indices and zero: a0 = 1, a-n = 1/an.
Worked examples (showing application)
Example 1
Simplify 25 × 2-2.
Use product rule: 25 + (-2) = 23 = 8.
Example 2
Simplify (x2 y)3.
Use power of product and power of power: x2×3 y3 = x6 y3.
Example 3 (scientific)
Express 450 000 in the form a × 10n where 1 ≤ a < 10.
450 000 = 4.5 × 100 000 = 4.5 × 105.
Example 4 (zero/negative)
Show that a0 = 1 (a ≠ 0).
From quotient rule: an ÷ an = an-n = a0. But an ÷ an = 1, so a0 = 1.
Exercises (in class / homework)
  1. Write each as an index:
    1. 4×4×4×4
    2. 10 000
    3. 0.001
  2. Simplify using laws of indices:
    1. 34 × 3-1
    2. 56 ÷ 52
    3. (23)4
    4. (6×7)2
  3. Algebra: Simplify:
    1. x5 ÷ x2
    2. (x2 y)3
    3. (a3 b-2)2
  4. Real-life: Write 3 200 000 in scientific notation. Convert 4.2 × 10-3 to ordinary form.
(Spend 10–20 minutes in class on items 1–2, then group work on 3–4.)
Suggested Answers
  1. a) 44 ; b) 104 ; c) 10-3
  2. a) 33 = 27 ; b) 54 = 625 ; c) 212 = 4096 ; d) 62 × 72 = 36 × 49 = 1764
  3. a) x3 ; b) x6 y3 ; c) a6 b-4
  4. 3 200 000 = 3.2 × 106. 4.2 × 10-3 = 0.0042.
Suggested learning experiences (classroom & home)
  • Start with physical activity: learners form groups and with cards show repeated multiplication (cards labelled 2, 2, 2 for 23) to link idea of "times" to index.
  • Use Kenyan examples: calculate area of a tomato bed 4 m × 4 m as 42 m2; estimate production growth using simple interest-like growth an for repeated multiplication (discuss small integer growth rates).
  • Practical calculator work: convert large/small populations or distances to scientific notation. Show how indices help when writing large numbers such as Kenya's national budget (simplified figures) or national population scaled estimates.
  • Group deduction task: give students concrete expansions (e.g., a×a×a × a×a) and ask them to derive the product and quotient rules — encourage writing the cancellation steps to see the quotient rule.
  • Worksheet + peer marking: short timed quiz on laws and conversions, then swap and discuss mistakes.
  • Extension: brief introduction to negative indices with money examples (1/2 of a1 shown as a-1), and to indices in computing (bytes as powers of 2).
Resources & tips for the teacher
  • Use a board to expand powers visibly and cancel factors when proving quotient rule.
  • Allow calculator checks for numeric answers but insist on showing index work for marks.
  • Local context: use examples from secondary school subjects (area, volume, finance) and simple biology (population doubling) to show applications.
Prepared for Essential Mathematics — Topic 1.0 Numbers & Algebra. Content suited to Kenyan syllabus style and learners aged ~15.

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