Simplifying Square Roots (Surds)

Introduction

• Simplifying square roots involves finding two numbers that multiply to create the number under the square root.
• One of these numbers should be a perfect square.

Process of Simplification

• Example 1: Simplify ( \sqrt{50} )

  • Identify square factors: 1, 4, 9, 16, 25, 36, etc.
  • 25 is a factor of 50, thus ( 50 = 25 \times 2 ).
  • Simplify: ( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} ).

• Example 2: Simplify ( \sqrt{96} )

  • Identify the largest square factor: 16.
  • ( 96 = 16 \times 6 ).
  • Simplify: ( \sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6} ).

Multiplying Surds

• Example: ( \sqrt{5} \times \sqrt{15} )

  • Multiply numbers under the roots: ( \sqrt{5 \times 15} = \sqrt{75} ).
  • Simplify: ( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} ).

• Example: ( \sqrt{10} \times \sqrt{12} )

  • Multiply to find: ( \sqrt{120} ).
  • Identify the largest square factor: 4.
  • Simplify: ( \sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30} ).

Dividing Surds

• Example: ( \frac{\sqrt{48}}{\sqrt{3}} )
  • Use division under a single root: ( \sqrt{\frac{48}{3}} = \sqrt{16} = 4 ).

Adding and Subtracting Surds

• Only possible if numbers under the roots are the same.

• Example: Add ( 5\sqrt{2} + 2\sqrt{2} )

  • Result: ( 7\sqrt{2} ).

• Example: Simplify ( 4\sqrt{10} + \sqrt{90} )

  • ( \sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10} ).
  • Combine: ( 4\sqrt{10} + 3\sqrt{10} = 7\sqrt{10} ).

Exam Problem Example

• 2014 Paper 1, Question 8: Simplify ( 4\sqrt{10} + \sqrt{90} )
  • ( \sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10} ).
  • ( \sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10} ).
  • Combine: ( 2\sqrt{10} + 4\sqrt{10} + 3\sqrt{10} = 9\sqrt{10} ).

Key Concepts

• Always look for the largest square factor.
• Multiply or divide inside the roots before simplifying.
• Only add or subtract surds with identical roots.
• Double-check the simplified result for further simplification opportunities.