IGCSE Math B 1.E Rationalising denominators Topic Practice

Without using a calculator and showing all your working, express
$$\frac{4-\sqrt{12}}{4+\sqrt{12}}$$
in the form $a-\sqrt b$ where a and b are integers.

Without using a calculator and showing all your working, express
$$\frac{11+\sqrt2}{3\sqrt2-1}$$
in the form $a+b\sqrt2$, where a and b are integers.

Without using a calculator and showing all your working, express
$$\frac{4-2\sqrt3}{\sqrt3+1}$$
in the form $a\sqrt3+b$, where a and b are integers.

Without using a calculator and showing all your working, express
$$\frac{4+2\sqrt7}{3-\sqrt7}$$
in the form $a+\sqrt b$, where a and b are integers.

Given that
$$p=\frac{1+\sqrt5}{2},$$
show that
$$\frac1p=p-1.$$
Show your working clearly.

Diagram NOT accurately drawn.

The diagram shows rectangle ABCD.
$$AD=BC=(2-\sqrt2)\text{ cm}$$$$\text{Area of }ABCD=3(5\sqrt2-2)\text{ cm}^2.$$
Show that the length of AB can be written in the form $(a+b\sqrt2)$ cm, where a and b are integers to be found.
Show your working clearly.

Without using a calculator, show that
$$\frac{(\sqrt{27}+\sqrt3)^2}{\sqrt{45}+3}$$
can be written in the form $a\sqrt b-a$, where a and b are integers.
Show your working clearly.

Answer parts (a) and (b).

Without using your calculator and showing all your working, express
$$\frac{6+2\sqrt3}{3\sqrt2-\sqrt8}$$
in the form $\sqrt p+\sqrt q$, where p and q are integers.