# 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: