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.

surdsSimplifying 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,…

Example 1: \sqrt{8}=\sqrt{4\times 2}=\sqrt{4}\times \sqrt{2}=2\sqrt{2}
The first step is to write 8 as a product of a square number and some other number.

Example 2: \sqrt{27}=\sqrt{9\times 3}=\sqrt{9}\times \sqrt{3}=3\sqrt{3}
Putting the square number first sometimes makes the process easier to remember.

Example 3: \sqrt{20}=\sqrt{4\times 5}=\sqrt{4}\times \sqrt{5}=2\sqrt{5}
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).

Example 4: \sqrt{32}=\sqrt{16\times 2}=\sqrt{16}\times \sqrt{2}=4\sqrt{2}
Next, make any simplifications.

Example 5: \sqrt{75}=\sqrt{25\times 3}=\sqrt{25}\times \sqrt{3}=5\sqrt{3}
Finally, remove the uneccessary multiplication sign.
Example 6: \sqrt{98}=\sqrt{49\times 2}=\sqrt{49}\times \sqrt{2}=7\sqrt{2}

Manipulating Surds

When manipulating or simplifying algebraic expressions involving surds it is useful to remember the following:
\sqrt{x}\times \sqrt{x}=x
x\times \sqrt{y}=\sqrt{y}\times x=x\sqrt{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}

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{5}{\sqrt{3}}=\frac{5}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{5\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{5\sqrt{3}}{9}
Example 2 – \frac{6}{\sqrt{8}}=\frac{6}{2\sqrt{2}}=\frac{3}{\sqrt{2}}=\frac{3}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{3\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{2}}{2}