As you make the vast step up to advanced study in maths, you will need a firm grasp on the fundamentals of simplifying surds.

Before you dive right into the nitty-gritty of worked examples, Beyond: Advanced can offer you the following resources (*great for teachers and students, alike*).

*For questions and answers on this topic, click here.*

*For a surds treasure hunt, with answers, click**here*.

### Simplifying Surds – Background

The name βsurdβ comes from the Latin *surdus*, which translates as βmuteβ or βunspokenβ. This used to apply to all irrational numbers but, now, it refers to a specific type of irrational number: those that can be written as the square root of an integer.

You will have come across Surds at GCSE; at A level it is important that you understand and can manipulate surds.

To simplify surds, there are two rules that you need to be able to use:

## Simplifying Surds – Example Content

**Example Question 1:**

Simplify the surd .

When simplifying surds, we need to find a factor that is a square number. In this case, we can see that 25 is a factor of 75, so we can write:

**Example Question 2:**

Simplify the surd .

The square root of a fraction can be calculated by finding the square root of both the denominator and numerator:

**Example Question 3:**

Find . Give your answer as a surd in its simplest form.

It is important to remember that . We can expand brackets that include surds in exactly the same way as we would expand algebraic brackets:

**Example Question 4:**

Express in the form where and are integers.

It is hard to see how to simplify , so letβs start by looking at . You may be able to spot that 150 is a multiple of 25, so we can write:

This then helps us simplify as we can assume that 6 is a factor:

Now, we can simplify :

When a surd is written on the denominator of a fraction, we *rationalise *the denominator to make it simpler. We do this by multiplying the numerator and denominator by an expression that will simplify the surd on the denominator.

**Example Question 5:**

Simplify .

In this case, we start by simplifying :

We now have:

We can rationalise the denominator by multiplying the numerator and denominator by .

We can do this because , which removes the root, and , so our multiplication does not affect the value of the fraction.

We could have rationalised this surd by multiplying the numerator and denominator by . However, by simplifying the surd first, we make sure our fraction is simplified and make the multiplication easier.

**Example Question 6:**

Simplify .

To do this, we consider . This is called the conjugate of ; it can be found by simply changing the sign of the surd. To help understand how this works, think back to the difference of two squares:

When we expand , we get:

We can now use this to remove the surd on the denominator:

So, to rationalise the denominator of a surd, we multiply the numerator and denominator by the conjugate of the denominator.

**Example Question 7:**

Simplify .

This time, our conjugate is ; we multiply our numerator and denominator by this. Since the numerator is also a surd, you may find it helpful to multiply them separately:

Bringing these together gives:

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