Laws of Indices— Negative & Fractional.

Q: What are the seven laws of indices in IGCSE Maths?

The seven laws are: (1) aᵐ × aⁿ = aᵐ⁺ⁿ, (2) aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (3) (aᵐ)ⁿ = aᵐⁿ, (4) a⁰ = 1, (5) a⁻ⁿ = 1/aⁿ, (6) a^(1/n) = the nth root of a, and (7) a^(m/n) = the nth root of aᵐ. These laws apply to all bases and are essential for simplifying algebraic expressions at Higher tier.

Q: What does a negative index mean?

A negative index means you take the reciprocal. For example, 2⁻³ = 1/2³ = 1/8. Think of it as "flip it". If the base is already a fraction, (2/3)⁻¹ = 3/2. This applies to both numerical and algebraic expressions in IGCSE Maths.

Q: How do you evaluate fractional indices?

A fractional index a^(m/n) means: take the nth root first (denominator), then raise to the power m (numerator). For example, 8^(2/3) means the cube root of 8 (which is 2), then squared (which is 4). The order can be reversed — root first or power first — but root first usually keeps numbers smaller.

Q: Are negative and fractional indices tested on both IGCSE tiers?

Negative and fractional indices are Higher tier content only. They do not appear on the Foundation tier papers. However, basic index laws (laws 1-4) are tested on both tiers. If your child is sitting Higher tier, mastering all seven laws is essential for grades 7-9.

All seven index laws explained with worked examples for IGCSE Higher tier grades 7-9.

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What are the laws of indices and how do negative and fractional powers work?

The seven laws of indices govern how powers (indices) behave when you multiply, divide, or raise them to further powers. A negative index means reciprocal: a⁻ⁿ = 1/aⁿ. A fractional index means root then power: a^(m/n) = ⁿ√(aᵐ). These laws are essential for simplifying expressions and solving equations at IGCSE Higher tier.

Tier: HigherTopic: Number / AlgebraPapers: 1 & 2

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

Key Takeaway

There are seven index laws but they all follow logical patterns. A negative index means reciprocal (flip it). A fractional index means root (denominator) then power (numerator). Master these two interpretations and the rest follows naturally.

What are the seven laws of indices?

The laws of indices (also called index laws or exponent rules) tell you how to handle powers when performing operations. Here are all seven laws that IGCSE Higher tier students must know:

Law 1 — Multiplication: a^m × aⁿ = a^m+n (same base: add the powers)
Law 2 — Division: a^m ÷ aⁿ = a^m−n (same base: subtract the powers)
Law 3 — Power of a power: (a^m)ⁿ = a^mn (multiply the powers)
Law 4 — Zero index: a⁰ = 1 (any non-zero base to the power 0 equals 1)
Law 5 — Negative index: a⁻ⁿ = 1/aⁿ (reciprocal)
Law 6 — Fractional index (unit): a^1/n = ⁿ√a (the nth root)
Law 7 — Fractional index (general): a^m/n = ⁿ√(a^m) (root then power, or power then root)

Laws 1-4 are tested on both Standard and Higher tiers. Laws 5-7 are Higher tier only and are among the most commonly tested topics for grades 7-9.

How do negative indices work?

A negative index means reciprocal — "flip it." The negative sign does not make the answer negative; it moves the base from numerator to denominator (or vice versa).

Key examples:

5⁻¹ = 1/5
2⁻³ = 1/2³ = 1/8
(3/4)⁻¹ = 4/3
(2/5)⁻² = (5/2)² = 25/4
x⁻ⁿ = 1/xⁿ (algebraic form)

The pattern is consistent: a negative index flips the position. If the term is in the numerator, it moves to the denominator, and vice versa. This is especially important in algebra when simplifying expressions like 3x⁻² = 3/x².

How do fractional indices work?

A fractional index combines a root and a power. The denominator of the fraction tells you which root to take, and the numerator tells you which power to raise to.

Key pattern: a^m/n = (ⁿ√a)^m

9^1/2 = √9 = 3
8^1/3 = ³√8 = 2
16^3/4 = (⁴√16)³ = 2³ = 8
27^2/3 = (³√27)² = 3² = 9
32^3/5 = (⁵√32)³ = 2³ = 8

Tip: Always take the root first (denominator), then apply the power (numerator). This keeps numbers small. For example, with 16^3/4, finding the fourth root of 16 first gives 2, then 2³ = 8. If you raised to the power 3 first (16³ = 4096), then took the fourth root, you would still get 8 — but the arithmetic is much harder.

Combining negative and fractional indices

When an index is both negative and fractional, apply both rules: flip (negative) and root-then-power (fractional). For example:

8^−2/3 = 1 / 8^2/3 = 1 / (³√8)² = 1 / 2² = 1/4

(25/9)^−1/2 = (9/25)^1/2 = √(9/25) = 3/5

Worked Examples

Grade 7

Simplify: 2x³ × 5x⁻¹

Step 1 (Multiply coefficients): 2 × 5 = 10

Step 2 (Add indices — Law 1): x³ × x⁻¹ = x³⁺⁽⁻¹⁾ = x²

Answer: 10x²

Grade 8

Evaluate: (16/81)^3/4

Step 1 (Apply the fraction to numerator and denominator): 16^3/4 / 81^3/4

Step 2 (Root first — fourth root): ⁴√16 = 2, ⁴√81 = 3

Step 3 (Then power — cube): 2³ = 8, 3³ = 27

Answer: 8/27

Grade 9

Simplify fully: (27x⁶)^(−2/3)

Step 1 (Apply the power to each factor): 27^−2/3 × (x⁶)^−2/3

Step 2 (Evaluate 27^−2/3): ³√27 = 3, then 3² = 9, then reciprocal → 1/9

Step 3 (Evaluate (x⁶)^−2/3): 6 × (−2/3) = −4, so x⁻⁴ = 1/x⁴

Answer: 1 / (9x⁴)

Common IGCSE Exam Mistakes

Thinking a negative index gives a negative answer: 2⁻³ = 1/8, not −8. The negative moves the base, it does not change the sign.

Confusing the root and power in fractional indices: In a^m/n, the denominator is the root and the numerator is the power — not the other way around.

Adding indices when the bases are different: 2³ × 3² cannot be simplified using Law 1. The bases must be the same.

Forgetting that a⁰ = 1: Students sometimes write a⁰ = 0 or a⁰ = a. Any non-zero number raised to the power 0 equals 1.

Exam Tips for Laws of Indices

Root first, power second: When evaluating fractional indices, always take the root (denominator) before the power (numerator) to keep numbers manageable.

Know your key roots: Memorise cube roots (8→2, 27→3, 64→4, 125→5), fourth roots (16→2, 81→3), and fifth root of 32→2. These appear repeatedly.

Write each step explicitly: Index questions carry 2-3 method marks. Show the law you are applying at each step to maximise marks even if the final answer is wrong.

Convert roots to index form for algebraic questions: Write √x as x^1/2 and ³√x as x^1/3 so you can apply index laws directly.

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