A Level Maths: Success With Surds
Introduction
Surds are also referred to as an irrational number. Sounds a little funny and a term that you may not have come across.
So what is an irrational number? To put it simply it is a number that cannot be written as a fraction.
A rational number is a number that can be written as a fraction such as the following:
\frac{3}{4}, 0.1=\frac{1}{10}, 3=\frac{3}{1}
So a surd such as \sqrt{2} cannot be written as a fraction and this is because the decimal version of such a number continues forever.
There are a few simple rules that surds follow such as:
\begin{aligned} & \sqrt{a} \times \sqrt{b}=\sqrt{a b} \\ & \sqrt{a} \times \sqrt{a}=a \\ & \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}} \end{aligned}Simplifying Surds
This would have been covered at GCSE level and even though your calculator can simplify surds for you, it is important that you are aware of the techniques that are involved.
Suppose you wanted to simply \sqrt{75}
In order to do this you need to think of the numbers that are a factor of 75 and if possible to think of a square number.
\sqrt{75}=\sqrt{5} \times \sqrt{15}=\sqrt{5} \times \sqrt{5} \times \sqrt{3}=5 \sqrt{3}Alternatively you could have said: \sqrt{75}=\sqrt{25} \times \sqrt{3}=5 \times \sqrt{3}=5 \sqrt{3}
Expanding Brackets With Surds
You will be familiar with expanding brackets as a part of algebra. You are also required to be able to expand brackets that contain surds and when using the rules mentioned above, this is not very difficult.
Expand \sqrt{2}(3+\sqrt{2})
As with algebra you multiply what is inside the bracket with what is outside the bracket. This will then give us the following result:
\begin{aligned} & 3 \sqrt{2}+\sqrt{2} \sqrt{2} \\ & =3 \sqrt{2}+2 \end{aligned}Expand (2+\sqrt{3})(4-\sqrt{3})
As with algebra, multiply each term together and then collect like terms to simplify your answer.
\begin{aligned} & =8-2 \sqrt{3}+4 \sqrt{3}-\sqrt{3} \sqrt{3} \\ & =8+2 \sqrt{3}-3 \\ & =5+2 \sqrt{3} \end{aligned}
Exam Style Question
\text { Simplify } \sqrt{75}-\sqrt{12} \text { giving your answer in the form } a \sqrt{3} \text {, where } a \text { is an integer. }First need to simply each of the surds, trying to think of that square number where possible:
\begin{aligned} & \sqrt{75}=\sqrt{25} \times \sqrt{3}=5 \sqrt{3} \\ & \sqrt{12}=\sqrt{4} \times \sqrt{3}=2 \sqrt{3} \end{aligned}Finally we then have: 5 \sqrt{3}-2 \sqrt{3}=3 \sqrt{3}
This question is worth 3 marks. The mistake that is made here is to simply input the values into a calculator. But for 3 marks you are expected to display your working.
Attending one of our intense classroom based A Level Maths Revision Courses you will be shown how to answer questions effectively in order to maximise your marks and therefore your final grade.
Rationalising The Denominator
Quite often you will see a fraction where the denominator contains a surd. It can be useful to rearrange the fraction so that the denominator becomes rational. This process is referred to as rationalising the denominator and is introduced at GCSE level and is also widely used at A Level.
Here are the rules that you need to follow:
Given:
\frac{1}{\sqrt{a}} \text { multiply by } \frac{\sqrt{a}}{\sqrt{a}}\frac{1}{a+\sqrt{b}} \text { multiply by } \frac{a-\sqrt{b}}{a-\sqrt{b}}
\frac{1}{a-\sqrt{b}} \text { multiply by } \frac{a+\sqrt{b}}{a+\sqrt{b}}
For the second and third rule you must change the sign of the original fraction in order for the rationalising to be successful. Doing this eliminates any surds in the denominator.
Examples
Rationalise the denominator for the following:
\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{7}}{7}\frac{1}{2+\sqrt{5}} \times \frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{2-\sqrt{5}}{4-2 \sqrt{5}+2 \sqrt{5}-5}=\frac{2-\sqrt{5}}{-1}=\sqrt{5}-2
\frac{2+\sqrt{3}}{3-\sqrt{11}} \times \frac{3+\sqrt{11}}{3+\sqrt{11}}=\frac{6+2 \sqrt{11}+3 \sqrt{3}+\sqrt{3} \sqrt{11}}{9+3 \sqrt{11}-3 \sqrt{11}-11}=\frac{6+2 \sqrt{11}+3 \sqrt{3}+\sqrt{33}}{-2}
The first and second questions, the answers look “normal” but not every answer is going to look like it should. You can check your answers to the first two using your calculator and you get what is seen here. But if you do it with the third question then your calculator gives you a long decimal. For this reason it is important not to be over reliant on your calculator and to be showing all your working out.
Exam Style Question
Given this exam question it is important to understand what is required and what steps you need to take. Given that it is a question regarding surds there is only so much you can do. In this case it should be clear that you need to rationalise the denominator.
\begin{aligned} &\frac{3-2 \sqrt{5}}{\sqrt{5}-1} \times \frac{\sqrt{5}+1}{\sqrt{5}+1}\\ &\frac{3 \sqrt{5}+3-10-2 \sqrt{5}}{5+\sqrt{5}-\sqrt{5}-1}\\ &\begin{aligned} & \frac{\sqrt{5}-7}{4} \\ & =-\frac{7}{4}+\frac{1}{4} \sqrt{5} \end{aligned} \end{aligned}Note how the final answer is written and what the question requires. When it comes to doing work with surds it is important not to just use your calculator but to show all your working out and to take your time reading the question.
As it goes, surds is not a difficult topic, but if you feel you need additional support then you should consider investing in an online A Level maths tutor. They will be able to monitor your progress on a regular basis and give other alternatives to dealing with questions which you may not have met whilst in your current school or college.
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