Simplify algebraic powers with integer exponents
Apply index laws to powers of the same variable · Simplify coefficients and exponents separately · Retain non-zero restrictions
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
- Integer exponent laws
- Algebraic factors
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 index laws to powers of the same variable
- Simplify coefficients and exponents separately
- Retain non-zero restrictions
Treat coefficients and bases separately
For non-zero x, xᵐxⁿ = xᵐ⁺ⁿ and xᵐ/xⁿ = xᵐ⁻ⁿ. Thus 8x⁵/(2x²) = 4x³ on x ≠ 0. Numerical coefficients are divided normally; their values do not become exponents.
A negative power indicates a reciprocal: q³q⁻⁴ = q⁻¹ = 1/q. For a power of a product, raise each factor: (2x³)² = 4x⁶.
Do not erase restrictions
A simplification is equivalent where the original expression is defined. x/x = 1 for x ≠ 0; the simplified constant does not make the original valid at zero. In Core use simple integer examples; fractional-variable exponents are a separate Extended lesson.
Sources for this lesson
Pause and explain
For q ≠ 0, what is q³q⁻⁴?
Worked example
Simplify 12a⁵/(3a²), stating the restriction.
Show the worked solution
- 12/3 = 4.
- a⁵/a² = a³ for a ≠ 0.
Answer 4a³, with a ≠ 0.
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.
Simplify x²x⁵ and x⁷/x³, stating restrictions for the quotient.
Give me a hint
Use the product and quotient laws separately.
Compare my reasoning
- Product = x⁷.
- Quotient = x⁴, with x ≠ 0.
x⁷; x⁴ for x ≠ 0.
Look for these in your work
- Select the right law
- Record the denominator restriction
Simplify (3a²)²/(9a), with a ≠ 0.
Give me a hint
Square the coefficient as well as the power.
Compare my reasoning
- Numerator = 9a⁴.
- Divide by 9a to get a³.
a³ for a ≠ 0.
Look for these in your work
- Raise every factor
- Subtract exponents in division
A student simplifies 5x⁻² to −5x². Correct the expression and evaluate it at x = 2.
Give me a hint
A negative exponent changes the location of the power.
Compare my reasoning
- 5x⁻² = 5/x², excluding x = 0.
- At x = 2, it gives 5/4.
5/x² for x ≠ 0; value 5/4 at x = 2.
Look for these in your work
- Use reciprocal notation
- Test the corrected form
Common mistakes
- Do not add exponents when dividing.
- Do not drop original denominator exclusions.
What can you explain now?
For q ≠ 0, what is q³q⁻⁴?
Compare with the explanation
1/q.
Add the exponents to get −1, then interpret the reciprocal.
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Make it stick
For q ≠ 0, what is q³q⁻⁴?
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.