### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Surds

Surds are irrational numbers that cannot be expressed as exact decimals or fractions. They are usually represented as square roots of non-perfect square numbers.

When working with surds, it’s important to simplify expressions by factoring out perfect squares and rationalising denominators when necessary. By doing this, surds can be presented in a simpler and more manageable form, making them easier to work with in mathematical problems.

Let’s look at how to simplify and rationalise surds.

## Simplifying Surds

To simplify a surd, we use the rule: .

Consider the following examples:

• • Now, let’s simplify an expression involving surds:  Ok, let’s look at some more examples.

Example 1:

Simplify    Example 2:

Simplify Start by expanding the expression: Now, multiply the terms: Combine like terms: The simplified expression is: Example 3:

Simplify We can use the difference of squares formula with: and Compute and :  Remember, Now apply the difference of squares formula: Multiply the terms: The simplified expression is: ## Rationalising Surds

When working with surds (square roots of non-perfect squares), sometimes we need to rationalise the denominator, which means removing the surds from the denominator. Let’s look at how to rationalise fractions containing surds in their denominators.

### Conjugates

The conjugate of is , and the conjugate of is . Conjugates are a useful tool for rationalising surds. To illustrate the process, let’s look at some examples.

Keep in mind this key formula when working with conjugates: It’s also important to remember this formula: Example 1:

Rationalise  Example 2:

Rationalise the following:

1. 2. 1.  2.  Example 3:

Rationalise To rationalise this fraction, multiply the numerator and denominator by the conjugate of , which is : Now apply the formula : Simplify the expression:  Example 4:

Rationalise: To rationalise this fraction, multiply the numerator and denominator by the conjugate of , which is : Now apply the formula : Simplify the expression:  Example 5:

Rationalise To rationalise the denominator, multiply the numerator and denominator by the conjugate of the denominator. This results in:     Simplified, we get: Example 6:

Rationalise Multiply by the conjugate of the denominator: This results in:  Simplified, we get: Example 7:

Rationalise Multiply by the conjugate of the denominator: This results in:   Simplify the expression:  Finally, we can simplify further by dividing both the numerator and the denominator by their highest common factor, which is 3: 