Surds Practice Questions

Write $\sqrt{162}$ in the form $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is as small as possible.

Simplify fully $\sqrt{8}\times\sqrt{6}$.

$\dfrac{12}{\sqrt{6}}$

$\dfrac{5}{2\sqrt{10}}$

Write $\sqrt{288}-\sqrt{50}+2\sqrt{98}$ in the form $a\sqrt{2}$, where $a$ is an integer.

Show clearly which is larger, $3\sqrt{7}$ or $5\sqrt{3}$.

Expand and simplify $(3-2\sqrt{5})^2$.

Give your answer in the form $a+b\sqrt{5}$, where $a$ and $b$ are integers.

Expand and simplify $(2\sqrt{3}+5)(\sqrt{3}-1)$.

Give your answer in the form $p+q\sqrt{3}$, where $p$ and $q$ are integers.

Work out the exact area of the triangle.

Give your answer in the form $a\sqrt{b}$ cm$^2$, where $a$ and $b$ are both integers greater than $1$.

Rationalise the denominator and simplify fully

$$\frac{4+\sqrt{3}}{2-\sqrt{3}}$$

Solve $\;\sqrt{8}\,x+3=\sqrt{50}+2x$

Give your answer in the form $\dfrac{a+b\sqrt{2}}{c}$, where $a$, $b$ and $c$ are integers.

Show that $(\sqrt{6}+\sqrt{2})^2=8+4\sqrt{3}$.

Hence express

$$\frac{(\sqrt{6}+\sqrt{2})^2}{\sqrt{3}-1}$$

in the form $a+b\sqrt{3}$, where $a$ and $b$ are integers.