Year 8 Mathematics · AC9M8N02

Exponent Laws

Build exponent laws from repeated multiplication, then apply product, quotient, power-of-a-power and zero-exponent rules only when their conditions are satisfied

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Key conceptTeach from the board

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 examplesWe do

Worked examples

AC9M8N02 - Exponent Laws
Example 1

Expand 3²×3⁴ and derive the product law.

Example 2

Simplify 7⁹÷7⁴ and explain cancellation.

Example 3

Use 4⁵÷4⁵ to explain 4⁰=1.

Example 4

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.

Curriculum examplesCopied content

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.

FrameworkMappingRelationship
Australian CurriculumAC9M8N02Canonical
VictoriaVC2M8N02Exact
NSW Stage 4MA4-IND-C-01; MAO-WM-01Exact
Questions and answersWith answers
  1. Why do exponents add in a product? The exponent counts equal-base factors, so factor counts combine.
  2. Why subtract in a quotient? Common factors cancel.
  3. Why is a⁰=1? aⁿ÷aⁿ=1 and the quotient law gives a⁰.
  4. Can different bases use these laws? Not unless first rewritten to a common base where valid.
Practice and reviewReady for practice
  • 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.
Curriculum alignmentStart here
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