Indices I (Copy)

IGCSE CORE Mathematics Cheat Sheet: Indices (Powers) and Laws of Indices

1. Understanding Indices (Exponents/Powers)

• Index (Exponent/Power): Tells how many times a number (base) is multiplied by itself.
  Example: 3⁴ = 3 × 3 × 3 × 3 = 81
• Types of Indices:
  • Positive Indices: Multiply the base by itself.
    Example: 5³ = 5 × 5 × 5 = 125
  • Zero Index:
    Any number raised to the power of 0 is 1 (except 0⁰).
    Example: 7⁰ = 1
  • Negative Indices:
    A negative power means reciprocal.
    Example: 2⁻³ = 1 / 2³ = 1/8

2. The Rules (Laws) of Indices

Note: These rules apply when the base is the same.

3. Key Examples

• Find the value of 7⁻²:
  7⁻² = 1 / 7² = 1 / 49
• Find the value of 2⁻³ × 2⁴:
  Use Product Rule: 2⁻³ × 2⁴ = 2¹ = 2
• Simplify (2³)²:
  Use Power of a Power: 2³ × 2³ = 2⁶ = 64
• Simplify 2³ ÷ 2⁴:
  Use Quotient Rule: 2³ ÷ 2⁴ = 2⁻¹ = 1/2

4. Working with Different Bases

• The index laws do not apply if bases are different:
  Example: 2³ × 3³ ≠ 6³
  You must calculate separately:
  2³ × 3³ = 8 × 27 = 216

5. Common Powers to Remember

6. Examples with Solutions

• Example 1:
  Simplify 3² × 3³
  = 3²⁺³ = 3⁵ = 243
• Example 2:
  Simplify (5²)³
  = 5²×³ = 5⁶ = 15625
• Example 3:
  Simplify 4⁵ ÷ 4²
  = 4³ = 64
• Example 4:
  Simplify 6⁰ + 2⁴
  = 1 + 16 = 17

7. Quick Reference Table

8. Tips for Exams

• Apply index rules only when bases are identical.
• A negative index means take the reciprocal.
• Always simplify expressions step by step using the correct law.
• Remember:
  Anything to the power 0 = 1.
  Don’t confuse a⁻² with −a² (the negative sign is different!).
• If indices involve fractions or decimals, stick to the same rules.