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 “surd” comes from the Latin surdus, which translates as “mute” or “unspoken”. 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 let’s 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:
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:
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:
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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