Simplify algebraic expressions with fractional exponents
Combine signed fractional exponents · Keep coefficients separate · State a positive-base domain for real root conventions
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Algebraic integer powers
- Fractional number powers
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
- Combine signed fractional exponents
- Keep coefficients separate
- State a positive-base domain for real root conventions
The same laws, stated conditions
For x > 0, fractional exponent laws follow the root definitions: x¹⁄²x³⁄² = x² and x⁵⁄³/x²⁄³ = x. Add or subtract the fractions carefully using a common denominator. Positive-base assumptions keep the expressions real and avoid zero denominators.
Coefficient and power operations
For (4x¹⁄²)/(2x⁻¹⁄²), divide coefficients to get 2 and subtract exponents to get x¹. A power of a product applies to every factor. For x > 0, (2x¹⁄²)² = 4x. Do not assume the laws extend without checking conditions to arbitrary negative real bases.
Sources for this lesson
Pause and explain
On x > 0, what is x¹⁄²x³⁄²?
Worked example
Simplify 3x¹⁄² × 2x⁻³⁄² on x > 0.
Show the worked solution
- Coefficients multiply to 6.
- Exponents add to −1, so the result is 6x⁻¹.
Answer 6/x, with x > 0.
Try a topic test
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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 x¹⁄³x²⁄³ for x > 0.
Give me a hint
Add the exponents.
Compare my reasoning
- 1/3+2/3 = 1.
- The result is x.
x, for x > 0.
Look for these in your work
- Use a common denominator
- Retain the stipulated domain
Simplify (6x³⁄²)/(3x⁻¹⁄²) for x > 0.
Give me a hint
Subtract a negative exponent.
Compare my reasoning
- Coefficients give 2.
- 3/2−(−1/2) = 2.
2x², for x > 0.
Look for these in your work
- Handle the negative sign
- Keep coefficients separate
Simplify (2x¹⁄²)², then check at x = 9. Why is a positive-base restriction useful?
Give me a hint
Square both factors.
Compare my reasoning
- The square gives 4x; at 9 it is 36.
- x > 0 ensures the root expressions and any related negative powers have consistent real meanings.
4x; value 36 at x = 9, on the stated positive domain.
Look for these in your work
- Raise every factor
- Explain the domain rather than discard it
Common mistakes
- Subtract the full signed exponent in a quotient.
- Root conventions require domain checks.
What can you explain now?
On x > 0, what is x¹⁄²x³⁄²?
Compare with the explanation
x².
The exponents add to 2.
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Make it stick
On x > 0, what is x¹⁄²x³⁄²?
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.