Indices II (Copy)

Indices Cheat Sheet (IGCSE Mathematics 0580)
Topic: Indices (Powers) – Positive, Zero, Negative, and Rules

1. Understanding and Using Indices

• Definition:
  An index (plural: indices) shows how many times a number is multiplied by itself.
• Examples:
  • 3² = 3 × 3 = 9
  • 5³ = 5 × 5 × 5 = 125
  • a² = a × a
• Positive Indices:
  Regular multiplication: x³ = x × x × x
• Zero Index Rule:
  • Any non-zero number raised to the power 0 is 1:
    • a⁰ = 1 (where a ≠ 0)
    • 5⁰ = 1
• Negative Indices:
  • A negative power means reciprocal:
    • a⁻ⁿ = 1 ÷ aⁿ
    • 2⁻³ = 1 ÷ (2³) = 1 ÷ 8
    • x⁻² = 1 ÷ (x²)

2. Rules of Indices

Examples

• Simplify (5x³)²
  • Apply power to 5 and x³ separately:
  • = 5² × (x³)²
  • = 25x⁶

• Simplify 12a⁵ ÷ 3a⁻²
  • Divide numbers and apply power rule:
  • = (12 ÷ 3) × (a⁵ ÷ a⁻²)
  • = 4a⁵⁻(–2)
  • = 4a⁷

• Simplify 6x⁷y⁴ × 5x⁻⁵y
  • Multiply numbers and apply index rules separately:
  • = (6 × 5)(x⁷ × x⁻⁵)(y⁴ × y¹)
  • = 30x⁷⁻⁵y⁴⁺¹
  • = 30x²y⁵

• Find value of x if 2ˣ = 32
  • Recognize 32 = 2⁵
  • So, 2ˣ = 2⁵
  • Therefore, x = 5

Special Cases to Remember

• Negative inside brackets:
  If (–3)², the answer is 9.
  But –3² (without brackets) means –(3²) = –9.
• Fractional indices (advanced cases not needed here):
  Only basic powers are required for 0580; fractional indices like 1/2 for roots are NOT tested at Core level.

Quick Reference Table

Key Exam Tip:
Always apply index laws step-by-step before doing basic arithmetic.
For negative powers, make reciprocal first if needed and then simplify.