Surds

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

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

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

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

$= 10\sqrt 2$