Surds

A surd is a square root which cannot be reduced to a rational number.

For example, \sqrt 4  = 2 is not a surd.

However \sqrt 5 is a surd.

If you use a calculator, you will see that \sqrt 5  = 2.236067977... and we will need to round the answer correct to a few decimal places. This makes it less accurate.

If it is left as \sqrt 5, then the answer has not been rounded, which keeps it exact.

Here are some general rules when simplifying expressions involving surds.

\sqrt a  \times \sqrt a  = a

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

\sqrt {\frac{a}{b}} = \frac{{\sqrt a }}{{\sqrt b }}

Now use the information above to try the example questions below.

Question

Simplify \sqrt {12}

= \sqrt {4 \times 3}

= \sqrt 4  \times \sqrt 3

= 2\sqrt 3

Question

Simplify \sqrt {48}

= \sqrt {16} \times \sqrt 3

= 4\sqrt3

Question

Simplify \sqrt {\frac{{16}}{9}}

= \frac{{\sqrt {16} }}{{\sqrt 9 }}

= \frac{4}{3}

Question

Simplify \sqrt 8  + \sqrt {18}  + \sqrt {50}

= \sqrt4 \sqrt 2  + \sqrt9 \sqrt 2  + \sqrt25 \sqrt 2

= 2\sqrt 2  + 3\sqrt 2  + 5\sqrt 2

= 10\sqrt 2