EDEXCEL GCSE HIGHER MATHS - What are Surds and Their Simplification

Edexcel GCSE Higher Maths: Surds and Their Simplification

What are Surds?

A surd is an irrational number that can be expressed as a root of an integer. It's essentially a way of representing numbers that cannot be simplified into a whole number or a simple fraction.

Examples of Surds:

• ?2
• ?5
• ?12

Simplifying Surds

The key to simplifying surds is finding the largest perfect square that divides the number inside the radical.

Steps for Simplifying Surds:

1. Find the largest perfect square that divides the number inside the radical.

2. For example, the largest perfect square that divides 12 is 4 (because 4 x 3 = 12).

3. Rewrite the radical as the product of the perfect square and the remaining factor.

4. In our example, ?12 = ?(4 x 3)

5. Simplify the square root of the perfect square.

6. ?(4 x 3) = ?4 x ?3 = 2?3

Important Points to Remember:

• You cannot simplify surds that involve variables unless the variable has a power that is a multiple of the index of the radical. For example, ?(x^2) can be simplified to x, but ?(x^3) cannot be simplified further.
• When simplifying surds, remember that the product of two surds is equal to the surd of their product. For example, ?2 x ?3 = ?(2 x 3) = ?6.

Practice Problems:

1. Simplify ?27
2. Simplify ?48
3. Simplify ?(125x^4)

Solutions:

1. ?27 = ?(9 x 3) = ?9 x ?3 = 3?3
2. ?48 = ?(16 x 3) = ?16 x ?3 = 4?3
3. ?(125x^4) = ?(25x^4 x 5) = ?(25x^4) x ?5 = 5x^2?5

Key takeaway: Understanding surds and their simplification is crucial for solving problems involving algebraic manipulations, quadratic equations, and trigonometry.