Simplifying Surds by Adding or Subtracting

Introduction

• Objective: Learn how to simplify surds by adding or subtracting.
• Square Numbers: Key to simplifying surds. List square numbers up to (12^2).
  • Examples: (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144).

Simplifying Surds

• Goal: Achieve square numbers under the square root sign, allowing simplification.
• Example: (\sqrt{4} = 2), (\sqrt{9} = 3).

Example Problems

Example 1: (\sqrt{48})

1. Break Down into Factors:
   • (48 = 4 \times 12)
   • Further simplify: (12 = 4 \times 3)
2. Separate Square Roots:
   • (\sqrt{4 \times 4 \times 3} = \sqrt{4} \times \sqrt{4} \times \sqrt{3})
3. Solve:
   • (\sqrt{4} = 2)
   • Final result: (2 \times 2 \times \sqrt{3} = 4\sqrt{3})

Example 2: (\sqrt{20} - \sqrt{5})

1. Break Down into Factors:
   • (20 = 4 \times 5)
2. Separate Square Roots:
   • (\sqrt{4} \times \sqrt{5} - \sqrt{5})
3. Solve:
   • (\sqrt{4} = 2)
   • Combine like terms: (2\sqrt{5} - 1\sqrt{5} = \sqrt{5})

Example 3: (2\sqrt{125} - 3\sqrt{45})

1. Break Down into Factors:
   • (125 = 5 \times 25)
   • (45 = 9 \times 5)
2. Separate Square Roots:
   • (2\sqrt{5 \times 25} - 3\sqrt{9 \times 5})
3. Solve:
   • (\sqrt{25} = 5) and (\sqrt{9} = 3)
   • Result: (10\sqrt{5} - 9\sqrt{5})
   • Combine like terms: (\sqrt{5})

Example 4: (\sqrt{36} + \sqrt{108} - \sqrt{25})

1. Break Down into Factors:
   • (108 = 9 \times 12)
   • (12 = 4 \times 3)
2. Simplify:
   • (\sqrt{36} = 6), (\sqrt{25} = 5)
   • Separate: (\sqrt{9 \times 12} = 3 \times \sqrt{4 \times 3})
3. Solve:
   • Result: (11 - 3\sqrt{3})

Example 5: (3\sqrt{49} + 2\sqrt{288} - \sqrt{72})

1. Break Down into Factors:
   • (288 = 144 \times 2)
   • (72 = 9 \times 8)
   • (8 = 4 \times 2)
2. Simplify:
   • (\sqrt{49} = 7), (\sqrt{144} = 12), (\sqrt{9} = 3)
3. Solve:
   • Result: (21 + 24\sqrt{2} - 12)
   • Combine like terms: (18\sqrt{2})

Tips for Simplification

• Always aim to express numbers as products of square numbers where possible.
• Look out for common factors and simplifications.
• Collect like surds to simplify further.