Use integer exponent laws and explain root exponents
Apply product, quotient and power laws · Interpret zero and negative exponents · Explain how a root extends integer exponent notation
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Multiplication and fractions
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Apply product, quotient and power laws
- Interpret zero and negative exponents
- Explain how a root extends integer exponent notation
Same-base laws
For a non-zero base a, aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. a⁰ = 1 and a⁻ⁿ = 1/aⁿ. A negative exponent does not mean a negative answer.
The conditions matter: 0⁻¹ is undefined, and 0⁰ is not assigned a value by these rules. Parentheses distinguish (−2)² = 4 from −2² = −4.
Extending the notation
For a positive base, defining a¹⁄ⁿ as the positive nth root makes (a¹⁄ⁿ)ⁿ = a. This explains root exponents as an extension of integer laws. Detailed fractional-exponent calculations are taught separately as Extended content; Core explains this connection and practises integer powers.
Sources for this lesson
Pause and explain
For non-zero a, what is a⁻³?
Worked example
Calculate 2⁴ × 2⁻³ and 4⁻².
Show the worked solution
- Equal bases give 2⁴⁻³ = 2.
- 4⁻² = 1/4² = 1/16.
Answer 2 and 1/16.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate 3² × 3⁻¹ and 5⁰.
Give me a hint
Use equal-base laws and the non-zero condition.
Compare my reasoning
- 3²⁻¹ = 3.
- 5⁰ = 1.
3 and 1.
Look for these in your work
- Use the correct law
- Check the base is non-zero
Calculate 2⁵ ÷ 2⁸ and (3²)³.
Give me a hint
Subtract exponents for division, multiply them for nested powers.
Compare my reasoning
- 2⁻³ = 1/8.
- 3⁶ = 729.
1/8 and 729.
Look for these in your work
- Retain a negative resulting exponent
- Keep parentheses
A student writes 0⁻¹ = 0 and (−2)² = −4. Correct both claims.
Give me a hint
Check division by zero and the squared base.
Compare my reasoning
- 0⁻¹ would require 1/0, so it is undefined.
- (−2)² = (−2)(−2) = 4.
0⁻¹ is undefined; (−2)² = 4.
Look for these in your work
- Check conditions
- Distinguish a base's sign from a leading minus
Common mistakes
- Add powers for a product; multiply them for a power of a power.
- Zero-base cases need separate checking.
What can you explain now?
For non-zero a, what is a⁻³?
Compare with the explanation
1/a³.
A negative exponent represents the reciprocal.
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Make it stick
For non-zero a, what is a⁻³?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.