Surds Practice Questions

Write $\sqrt{45}$ in the form $a\sqrt{5}$, where $a$ is an integer.

Simplify fully $\sqrt{27}+\sqrt{48}$.

Simplify fully $\sqrt{6}\times\sqrt{15}$.

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

Rationalise the denominator of $\dfrac{9}{\sqrt{6}}$.

Give your answer in its simplest form.

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

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

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

Find Priya's mistake.

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

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

Show that the area of the triangle is exactly $12$ cm$^2$.

A rectangle has the same area as the triangle.

The width of the rectangle is $\sqrt{8}$ cm.

Work out the length of the rectangle. Give your answer in the form $a\sqrt{2}$ cm, where $a$ is an integer.

Show that $\dfrac{12}{5+\sqrt{7}}$ can be written in the form $\dfrac{a+b\sqrt{7}}{3}$, where $a$ and $b$ are integers.

Show that $\dfrac{\sqrt{45}-\sqrt{5}}{3+\sqrt{5}}$ can be written in the form $\dfrac{a+b\sqrt{5}}{2}$, where $a$ and $b$ are integers.

Show that the length of the rectangle is $(5+\sqrt{5})$ cm.

Work out the exact perimeter of the rectangle.

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