OCR GCSE Maths: Introduction to Surds

Introduction to Surds

Surds are numbers that cannot be expressed as a simple fraction. They are essentially square roots of numbers that are not perfect squares.

Key Concepts:

• Simplifying Surds: The goal is to extract any perfect square factors from the number under the radical sign (?).
• Operations with Surds: You can add, subtract, multiply, and divide surds, following the rules of arithmetic.
• Rationalizing Denominators: This involves getting rid of any surds in the denominator of a fraction by multiplying both the numerator and denominator by a suitable factor.

Let's explore these concepts with some examples:

1. Simplifying Surds:

• Example: Simplify ?72
  • Find the largest perfect square that divides 72 (which is 36).
  • Rewrite ?72 as ?(36 × 2)
  • Use the property ?(a × b) = ?a × ?b to get ?36 × ?2 = 6?2

2. Operations with Surds:

• Addition and Subtraction: Only surds with the same number under the radical sign can be added or subtracted.
  • Example: 3?5 + 2?5 = (3+2)?5 = 5?5
• Multiplication and Division:
  • Example: ?3 × ?12 = ?(3 × 12) = ?36 = 6
  • Example: ?24 / ?6 = ?(24/6) = ?4 = 2

3. Rationalizing Denominators:

• Example: Simplify 1/?3
  • Multiply both numerator and denominator by ?3
  • (1 × ?3) / (?3 × ?3) = ?3 / 3

Key Points:

• Exact Calculations: Surds are used to represent exact values, avoiding the need for approximations using calculators.
• Understanding the Concepts: Mastering these concepts will help you solve more complex problems involving surds.

Practice Makes Perfect!

To become proficient with surds, practice solving various problems involving simplification, arithmetic operations, and rationalizing denominators. You'll find that working with surds becomes increasingly intuitive with regular practice.