Surds are irrational numbers left in their root form. Most of the time this is in the form \sqrt{x}, although occasionally it can be \sqrt[3]{x}.

The reason for this is that it is much easier to write numbers in this form, as opposed to decimals that continue forever.

For example, to answer the equation, x^2 = 2.

Our answers could either be \pm \sqrt{2} or \pm 1.414213562…, which do you think is simpler? You will encounter Surds a lot when solving Quadratic Equations.

Did you know? – The story goes that when one of Pythagoras’ students discovered that \sqrt{2} was an irrational number, Pythagoras ordered for him to be drowned!

Operations with Surds

Multiplication of Surds

In general:

\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}

Basically, if you have two surds multiplied together, then you can combine the two by multiplying together the two numbers under the square root sign.

A few more examples:

\sqrt{5} \times \sqrt{7} = \sqrt{35}

As 35 is not a square number, this cannot be simplified any more.

\sqrt{12} \times \sqrt{15} = \sqrt{12 \times 15} = \sqrt{180}

You may notice that we can simplify the last answer even more.

\sqrt{180} = \sqrt{4} \times \sqrt{5} \times \sqrt{9} = 2 \times \sqrt{5} \times 3 = 6\sqrt{5}

It’s important to notice here that if the number under the square root sign contains a factor that is a square number, then we can simplify the surd by taking the square root of this factor outside of the square root sign.

A more complicated example.

(1- \sqrt{2})(2-\sqrt{8})

Although this may look complicated, we multiply this as we would any other set of brackets.

(1- \sqrt{2})(2-\sqrt{8})
2-\sqrt{8} -2\sqrt{2} + \sqrt{2}\sqrt{8}
2-2\sqrt{2} -2\sqrt{2} + \sqrt{16} (Note that \sqrt{8} = 2\sqrt{2})
2-4\sqrt{2} + 4

Addition and Subtraction of Surds

When adding and subtracting surds we can treat them as if we were adding xs and ys. If the surds are the same then we can group them, if not then there is nothing we can do.

For example – 23\sqrt{5} - 13\sqrt{5} = 10\sqrt{5}

However, 23\sqrt{5} - 13\sqrt{7} = 23\sqrt{5} - 13\sqrt{7} cannot be simplified because the number under the square roots are different, (\sqrt{5} and \sqrt{7} ).

Rationalising the Denominator

Mathematicians like things to be neat, and fractions containing a surd denominator are considered untidy. This little problem can be solved by multiplying the top and the bottom of the fraction by a particular expression, removing the surd from the bottom of a fraction.

Fractions in the form \frac{a}{\sqrt{b}}, multiply both top and bottom by \sqrt{b}

Example – Rationalise the fraction \frac{1}{\sqrt{5}}

\frac{1 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{\sqrt{5}}{5}

A more complicated example

Fractions in the form \frac{a}{b + \sqrt{c}}, multiply top and bottom by b - \sqrt{c}

Example – Rationalise the fraction \frac{3}{4 + \sqrt{3}}

In these example you change the sign in front of the surd in the denominator of the fraction, doing so removes the square root from the bottom of your fraction.

\frac{3(4 - \sqrt{3})}{(4 + \sqrt{3})(4 - \sqrt{3})}
\frac{3(4 - \sqrt{3})}{16 +4\sqrt{3}- 4\sqrt{3} -3}
\frac{3(4 - \sqrt{3})}{13} or \frac{12 - 3\sqrt{3}}{13}