Simplifying Surds

Introduction

• Simplifying surds involves rewriting a surd in its simplest form so that the radicand (number under the root) has no square factors.
• Surds are simplified using three laws derived from the laws of indices:
  • (\sqrt{m} \times \sqrt{n} = \sqrt{mn})
  • (\sqrt{m} \div \sqrt{n} = \sqrt{\frac{m}{n}})
  • (\sqrt{m} \times \sqrt{m} = m)

How to Simplify Surds

1. Find a square number factor: Identify a square number that is a factor of the radicand.
2. Rewrite and evaluate: Express the surd as a product of the square number and another factor, then evaluate the square root of the square number.
3. Repeat if necessary: Continue simplifying if the radicand still has square factors.

Examples

Example 1: (\sqrt{8})

• Step 1: 4 is a factor of 8.
• Step 2: (\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \times \sqrt{2})
• Final answer: (\sqrt{8} = 2\sqrt{2})

Example 2: (\sqrt{45})

• Step 1: 9 is a factor of 45.
• Step 2: (\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3 \times \sqrt{5})
• Final answer: (\sqrt{45} = 3\sqrt{5})

Example 3: (\sqrt{240})

• Step 1: 4 is a factor of 240.
• Step 2: (\sqrt{240} = \sqrt{4 \times 60} = 2 \times \sqrt{60})
• Step 3: Simplify (\sqrt{60}) as (2\sqrt{15})
• Final answer: (\sqrt{240} = 4\sqrt{15})

Example 4: (\sqrt{108})

• Step 1: 9 is a factor of 108.
• Step 2: (\sqrt{108} = \sqrt{9 \times 12} = 3 \times \sqrt{12})
• Step 3: (\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3})
• Final answer: (\sqrt{108} = 6\sqrt{3})

Example 5: (\sqrt{720})

• Step 1: 16 is a factor of 720.
• Step 2: (\sqrt{720} = \sqrt{16 \times 45} = 4 \times \sqrt{45})
• Step 3: (\sqrt{45} = 3 \sqrt{5})
• Final answer: (\sqrt{720} = 12\sqrt{5})

Prime Factor Tree Method

• An alternative method for simplifying surds, especially for large numbers.
• Break down the radicand into its prime factors and simplify.
• Example: (\sqrt{720}) involves breaking down into prime factors and simplifying multiple times.

Common Misconceptions

• Incorrectly rewriting the radicand without one factor being a square number.
• Failing to simplify fully by missing square factors.

Practice Questions

1. Simplify (\sqrt{84}) to (2\sqrt{21})
2. Simplify fully (\sqrt{540}) to (6\sqrt{15})
3. Given (\sqrt{912} = 4\sqrt{k}), find (k) (Answer: 57)
4. Simplify fully (5\sqrt{80}) to (20\sqrt{5})

Learning Outcomes

• Identify a surd as an irrational root
• Ability to simplify surds effectively

Resources

• Worksheets and further guidance on simplifying surds for GCSE Maths are available for further practice.