Surds are square roots which can’t be reduced to rational numbers. Some can be simplified using various rules or by rationalising the denominator.

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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 \(\sqrt 8 + \sqrt {18} + \sqrt {50}\)

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

\[= 2\sqrt 2 + 3\sqrt 2 + 5\sqrt 2\]

\[= 10\sqrt 2\]