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Indices and Powers for IGCSE Maths

Laws of indices including negative and fractional powers. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding indices and po

What You Need to Know

Laws of indices including negative and fractional powers. This subtopic is part of Algebra in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding indices and powers is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Indices and Powers

The laws of indices (indices = powers/exponents) govern how to simplify expressions with powers. The key laws are: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a. Fractional indices combine roots and powers: a^(m/n) = (ⁿ√a)ᵐ. These appear throughout IGCSE 0580 — in number, algebra, and sometimes in sequences questions.

Step-by-Step Method

  1. 1

    Identify which law to use

    Same base × same base → add indices. Same base ÷ same base → subtract indices. Power raised to power → multiply indices.

  2. 2

    Handle zero and negative indices

    a⁰ = 1 for any non-zero a. a⁻ⁿ = 1/aⁿ means 'flip it': 3⁻² = 1/9.

  3. 3

    Handle fractional indices

    a^(1/n) = ⁿ√a. a^(m/n) = (ⁿ√a)ᵐ. Evaluate the root first (denominator of the fraction), then apply the power (numerator).

  4. 4

    Apply to algebraic expressions

    (2x³)⁴ = 2⁴ × x¹² = 16x¹². Each factor inside the bracket gets the outer power applied to it.

  5. 5

    Solve index equations

    Write both sides with the same base, then equate the indices: 2^(3x) = 8 → 2^(3x) = 2³ → 3x = 3 → x = 1.

Worked Example

Question

Evaluate: (a) 27^(2/3) (b) (4/9)^(−1/2)

Solution

Part (a): 27^(2/3) Denominator 3 means cube root: ³√27 = 3. Numerator 2 means square: 3² = 9. 27^(2/3) = 9 Part (b): (4/9)^(−1/2) Negative power means flip the fraction: (9/4)^(1/2) Power 1/2 means square root: √(9/4) = 3/2 (4/9)^(−1/2) = 3/2 Answers: (a) 9 (b) 3/2

Exam Tips for Indices and Powers

  • For a^(m/n): root first (denominator n), then power (numerator m) — doing it in reverse is harder and more error-prone.
  • a⁰ = 1, NOT 0. This catches many students out: 7⁰ = 1.
  • Negative index does NOT make the result negative: 3⁻² = 1/9, not −9.
  • When simplifying algebra with indices, apply the outer power to EVERY factor inside the bracket: (2x³)⁴ = 16x¹², not 2x¹².

Practice Questions

Q1: Simplify: (a) x⁵ × x³ (b) (y⁴)³ ÷ y⁶ (c) (3a²b)³

Show hint

(a) Add indices: x⁸. (b) Power rule then subtract: y¹²÷y⁶=y⁶. (c) 27a⁶b³.

Q2: Evaluate 64^(−2/3).

Show hint

Cube root of 64 = 4. Then 4² = 16. Negative power flips it: 1/16.

Q3: Solve 5^(x+1) = 125.

Show hint

125 = 5³. So x + 1 = 3, x = 2.

Frequently Asked Questions

What is indices and powers in IGCSE Maths?

Laws of indices including negative and fractional powers.

Is indices and powers in the Core or Extended syllabus?

Indices and Powers is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise indices and powers effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on indices and powers rather than trying to cover everything at once.

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