Apply integer and fractional index laws
Combine powers with common bases, interpret zero and negative indices, and use fractional indices for Extended work.
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 2⁻³?
What you will learn
- Apply product, quotient and power-of-a-power laws
- Interpret zero and negative integer powers
- Connect fractional indices to roots for Extended
Combine powers only when bases match
For a ≠ 0, aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ. Also (ab)ⁿ = aⁿbⁿ. Do not apply these rules across a sum.
a⁰ = 1 for non-zero a; a⁻ⁿ = 1/aⁿ. A negative index means a reciprocal, not a negative value.
Fractional powers describe roots
For a positive, a^(1/n) is its nth root and a^(m/n) = (ⁿ√a)ᵐ where the real root exists. For example, 16^(3/4) = (⁴√16)³ = 8.
Core uses integer indices in this section; fractional indices are Extended content. Record non-zero restrictions when dividing by powers.
Pause and explain
What is 16^(1/2)?
Worked example
Simplify (x³ × x⁻²)² and evaluate 81^(3/4).
Show the worked solution
- Inside, x³ × x⁻² = x for x ≠ 0.
- Squaring gives x².
- The fourth root of 81 is 3, and 3³ = 27.
Answer x² for x ≠ 0; 81^(3/4) = 27.
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 (a⁴ × a⁻¹)/a² and evaluate 32^(2/5).
Give me a hint
Combine common-base exponents; interpret a fifth root before squaring.
Compare my reasoning
- The exponent is 4 − 1 − 2 = 1, giving a.
- The fifth root of 32 is 2.
- 2² = 4.
a; 4.
Look for these in your work
- Exponents combined correctly
- Common base retained
- Fractional root evaluated
Simplify (3x²)²/(9x) and evaluate 64^(2/3).
Give me a hint
Square both numerator factors; take the cube root before squaring.
Compare my reasoning
- (3x²)² = 9x⁴.
- Divide by 9x to get x³, for x ≠ 0.
- The cube root of 64 is 4; 4² = 16.
x³ for x ≠ 0; 16.
Look for these in your work
- Coefficient squared
- Powers reduced
- Root and power order understood
A cube has volume 125 cm³. Find its side and surface area, expressing the area using a fractional power of 125.
Give me a hint
Take the cube root for the side, then square the side and multiply by six.
Compare my reasoning
- Side = 125^(1/3) = 5 cm.
- Surface area = 6 × 5² = 150 cm².
- Since 5² = 125^(2/3), area = 6 × 125^(2/3) cm².
Side 5 cm; surface area 150 cm² = 6 × 125^(2/3) cm².
Look for these in your work
- Cube root gives a length
- Area units squared
- Power relation checked
Common mistakes
- Combining unlike bases
- Reading a negative index as a negative answer
- Applying laws across addition
- Using fractional-index methods for Core-only content
What can you explain now?
Simplify 5³/5⁵.
Compare with the explanation
1/25
5^(3−5) = 5⁻² = 1/25.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
When can powers combine, what does a negative index mean, and how does a fractional index encode a root?
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.