Surds are essentially square roots of numbers that are not square. For example, 16 is a square number, if you root it you get 4. 8 is not a square number, if you root it you get \sqrt{8}. This is an example of a surd.

Simplifying Surds

The trick to simplifying surds is to consider the number within the square root and see if you can identify any square factors of this number. Square factors are numbers that you can divide by (and obtain an integer result) that happen to be square numbers: 1,4,9,16,25,…

Take \sqrt{8} to illustrate how to simplify a surd:

\sqrt{8}=\sqrt{4\times 2}=\sqrt{4}\times \sqrt{2}=2\times\sqrt{2}=2\sqrt{2}

The steps are given by the following:

  1. The first step is to write 8 as a product of a square number and some other number.
  2. Putting the square number first sometimes makes the process easier to remember.
  3. Subsequently, the root can be split out into two individual roots. Note that this is only true for multiplication (and division) and not addition (or subtraction).
  4. Next, make any simplifications.
  5. Finally, remove the uneccessary multiplication sign.

Example 1

\sqrt{27}=\sqrt{9\times 3}=\sqrt{9}\times \sqrt{3}=3\times\sqrt{3}=3\sqrt{3}

Example 2

\sqrt{20}=\sqrt{4\times 5}=\sqrt{4}\times \sqrt{5}=2\times\sqrt{5}=2\sqrt{5}

Example 3

\sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\times \sqrt{2}=4\times\sqrt{2}=4\sqrt{2}

Example 4

\sqrt{75}=\sqrt{25\times 3}=\sqrt{25}\times \sqrt{3}=5\times\sqrt{3}=5\sqrt{3}

Example 5

\sqrt{98}=\sqrt{49\times 2}=\sqrt{49}\times \sqrt{2}=7\times\sqrt{2}=7\sqrt{2}


Manipulating Surds

When manipulating or simplifying algebraic expressions involving surds it is useful to remember the following:

\left(\sqrt{x}\right)^2=\sqrt{x}\times \sqrt{x}=x

x\times \sqrt{y}=\sqrt{y}\times x=x\sqrt{y}

\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)=x-y

Example 1

\left(\sqrt{3}+2\right)^2=\left(\sqrt{3}+2\right)\left(\sqrt{3}+2\right)=\sqrt{3}\times\sqrt{3}+2\times\sqrt{3}+\sqrt{3}\times 2 +2\times 2=3+2\sqrt{3}+2\sqrt{3}+4=7+4\sqrt{3}

Example 2

(7-\sqrt{5})(4+3\sqrt{5})=28-4\sqrt{5}+21\sqrt{5}-5=23+17\sqrt{3}

Example 3

surds


Rationalising the Denominator

Recall that the denominator is the bottom of a fraction. It is possible to have a surd in the denominator of a fraction. Rationalising the denominator is where you remove the surd from the denominator. Consequently, this could mean that the surd appears on the top of the fraction i.e. in the numerator. This is OK as long as they don’t appear on the bottom. Therefore, the idea is to find an equivalent fraction where there is no surd in the denominator. Unsurprisingly, this can be achieved by multiplying or dividing the top and bottom by the same number. Inspecting the bottom of the fraction will tell what number to use.

Example 1

\frac{6}{\sqrt{3}}=\frac{6}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{6\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{6\sqrt{3}}{3}=2\sqrt{3}

Since the denominator is root 3, this is what we should multiply the top and bottom. It turns the bottom into a simple 3.

Example 2

\frac{8}{\sqrt{8}+2}=\frac{8}{2\sqrt{2}+2}=\frac{4}{\sqrt{2}+1}=\frac{4}{\sqrt{2}+1}\times\frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{4\sqrt{2}-4}{2-\sqrt{2}+\sqrt{2}-1}=\frac{4\sqrt{2}-4}{1}=4\sqrt{2}-4

Note that we simplified the surd then the fraction in the first two steps. It is then possible to rationalise the denominator by multiplying top and bottom by the denominator with the sign changed. As you can see from this example, it causes the surd to cancel.

Example 3

\frac{7}{\sqrt{2}-\sqrt{5}}=\frac{7}{\sqrt{2}-\sqrt{5}}\times\frac{\sqrt{2}+\sqrt{5}}{\sqrt{2}+\sqrt{5}}=\frac{7\sqrt{2}+7\sqrt{5}}{2+\sqrt{2}\sqrt{5}-\sqrt{2}\sqrt{5}-5}=-\frac{7\sqrt{2}+7\sqrt{5}}{3}

The technique for rationalising the denominator in Example 3 is very similar to that in Example 2.



Past Surds Exam Questions

We have collated past exam questions on Surds so that you may focus your concentration on this particular subject (answers on the back pages). Visit our Questions by Topic page to see these and other topics you can focus on.

Go to Questions by Topic



Pure Maths Practice Papers

Are you ready to test your knowledge on all of the Pure Maths topics so far? If so…

Go to Practice Papers

Also see the THINGS TO REMEMBER page.

If you’d like to further your knowledge, why not look up the proof that root 2 is irrational.