Product
2³×2⁴=(2×2×2)(2×2×2×2)=2⁷.
Learning goals
- Explain exponent notation as repeated multiplication.
- Derive product, quotient and power-of-a-power laws from expanded form.
- Explain why a⁰=1 for non-zero a.
- Apply and combine exponent laws accurately.
Prerequisite knowledge
Recall factors, powers, square/cube numbers, multiplication/division and order of operations. Be able to expand 2⁴ as 2×2×2×2.
Key concept
Exponent laws are shortcuts for counting repeated factors. If the same base is multiplied, factor counts add: aᵐ×aⁿ=aᵐ⁺ⁿ. If equal-base powers are divided, common factors cancel: aᵐ÷aⁿ=aᵐ⁻ⁿ for a≠0.
A power raised to a power repeats the entire factor group, so (aᵐ)ⁿ=aᵐⁿ. The zero exponent follows from consistency: a³÷a³=1 and also a³⁻³=a⁰, hence a⁰=1 for a≠0.
Always check the operation and base before choosing a law. The laws do not turn ordinary addition into exponent arithmetic, and equal-base conditions matter.
Worked examples
aᵐ × aⁿ→m+n factors of a→aᵐ⁺ⁿ
(aᵐ)ⁿ→n groups of m factors→aᵐⁿ
Quotient and zero exponent
5⁶÷5²=5⁴. Also 7⁴÷7⁴=7⁰=1.
Power of a power
(3²)⁴ has four groups of two factors, so 3⁸.
Combined laws
(2³×2⁵)²÷2⁴=(2⁸)²÷2⁴=2¹⁶÷2⁴=2¹².
Common misconceptions
- Add exponents whenever powers are added. Product/quotient laws concern multiplication/division, not 2³+2⁴.
- Multiply exponents in a product. Multiplication of exponents belongs to (aᵐ)ⁿ.
- Any number to power 0 is 0. For non-zero a, a⁰=1.
- Different bases can be merged. 2³×3² is not 6⁵.
Guided practice
- Expand 3²×3⁴ and derive the product law.
- Simplify 7⁹÷7⁴ and explain cancellation.
- Simplify (5³)².
- Use 4⁵÷4⁵ to explain 4⁰=1.
Independent practice
- Simplify 2⁵×2³.
- Simplify 10⁷÷10².
- Simplify (3⁴)³.
- Evaluate 9⁰ and state the base condition.
- Simplify 5²×5⁶÷5³.
- Explain why 2³+2⁴ cannot become 2⁷.
- Correct the error (4²)³=4⁵.
Reasoning/problem-solving
A student simplifies (2³×2⁴)² as 2⁹ because 3+4+2=9. Diagnose the error and show two correct methods that agree.
Questions and answers
- Why do exponents add in a product? The exponent counts equal-base factors, so factor counts combine.
- Why subtract in a quotient? Common factors cancel.
- Why is a⁰=1? aⁿ÷aⁿ=1 and the quotient law gives a⁰.
- Can different bases use these laws? Not unless first rewritten to a common base where valid.
Practice and review
- Simplify 3⁴×3⁷÷3⁵ and justify each exponent change.Track equal-base factors.
- Evaluate (2³)⁴÷2⁵.Use power-of-a-power, then quotient.
- Explain why 6⁰=1 using a quotient identity.Start with 6³÷6³.
Check understanding
- I derive laws from factors.
- I check bases and operations.
- I distinguish the main exponent laws.
- I justify the zero exponent.
Exit ticket: Why is (a³)²=a⁶ rather than a⁵?
Teacher + parent guidance
Teacher
Require expanded-factor justification before shortcuts. Contrast near-miss forms such as 2³×2⁴, 2³+2⁴ and (2³)⁴.
Parent/carer
Ask what the exponent counts and have the learner expand one expression to verify a rule.
Support: use small numerical bases, expanded repeated factors and a scaffolded law table.
Core: choose among product, quotient, power-of-a-power and zero laws, then verify with expansion.
Extend: derive a⁰=1, combine multiple laws and diagnose expressions where a law’s conditions fail.
Core: choose among product, quotient, power-of-a-power and zero laws, then verify with expansion.
Extend: derive a⁰=1, combine multiple laws and diagnose expressions where a law’s conditions fail.
Curriculum alignment
Australian Curriculum v9.0 — AC9M8N02: establish and apply exponent laws with positive integer exponents and the zero exponent.
Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8N02: Exact.
NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-IND-C-01; MAO-WM-01: Exact.
| Framework | Mapping | Relationship |
|---|---|---|
| Australian Curriculum | AC9M8N02 | Canonical |
| Victoria | VC2M8N02 | Exact |
| NSW Stage 4 | MA4-IND-C-01; MAO-WM-01 | Exact |
Practice/teaching resources
Official curriculum references
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Products and Exponents Raised to an Exponent
Khan Academy — Reason about powers of products and powers of powers by expanding repeated multiplication. The zero-exponent rule remains in the written topic.
As you watch: Why do you multiply exponents in a power of a power?
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Try it: Expand (2 cubed) squared as repeated multiplication, then express it as one power. Compare it with 2 cubed times 2 squared.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Establish and apply the exponent laws with positive integer exponents...
Mapped skill: establish and apply the exponent laws with positive integer exponents and the zero-exponent, using exponent notation with numbers
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M8N02 · Year 8 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M8N02 · Level 8 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-IND-C-01 + MAO-WM-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 8 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 8 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 9, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 8 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M8N02 — Exponent Laws — AC9M8N02
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