Indices
An index describes a power. First identify the operation and the base: multiplying equal bases adds indices, while raising a power to another power multiplies them. Addition is a different operation and does not obey either rule.
What this guide covers
Positive, zero, negative and fractional indices and their laws. For the fractional-power examples here, bases are positive; zero cannot be raised to a negative power.
Syllabus reference: 2026 Mathematics 4052, N1, page 5. These are Mentora search groupings, not new official syllabus categories.
The content in this guide was also checked against 2027 G3 Mathematics K310, N1, page 5 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.
Same symbols, different operations
- Multiply: a² × a³ = a⁵
- Add the indices because the base is the same.
- Power: (a²)³ = a⁶
- Multiply the indices: three factors of a².
- Add: a² + a³
- No index law combines these unlike terms. At a = 2 the sum is 12, not 2⁵ = 32.
- Reciprocal: a⁻² = 1/a²
- A negative index does not make the value negative. Require a ≠ 0.
Worked example
Simplify (2a³)² ÷ (4a), where a ≠ 0.
Square both factors: (2a³)² = 4a⁶. Then 4a⁶/(4a) = a⁵. The coefficient is squared too; the restriction remains because the original divisor is zero at a = 0.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Evaluate 272/3.
Show worked answer
Take the cube root then square: 3² = 9.
Check 2
Simplify x⁰ + x⁻¹, where x ≠ 0.
Show worked answer
x⁰ = 1 and x⁻¹ = 1/x, giving 1 + 1/x. Adding indices here would be invalid.
Check 3
Is (a + b)² = a² + b² always true? Use a = 2, b = 3 to check.
Show worked answer
The left side is 25; the right side is 13. Expansion gives a² + 2ab + b², so the missing cross term is 2ab = 12.
Mistakes to watch for
- Check that the bases are identical before adding indices in a product.
- Apply an outside power to the coefficient as well as the variable.
- Zero index gives 1 for a non-zero base, not 0; do not use this to assign a value to 0⁰.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
Useful foundations to check
These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.
Applications and next topics
Index laws help simplify algebraic fractions and interpret power graphs. Expansion shows why a power of a sum cannot be split into separate powers.
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Check power and expansion errors.
Sources and review
Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-12; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.