Surds: Practice Questions

Rationalise the denominator of the expression: $$\frac{18}{\sqrt{6}}$$

Rationalise the denominator and simplify the expression:
\( \frac{10}{\sqrt{7} - \sqrt{2}} \)

Simplify the surd expression: $$\sqrt{72}$$

Rationalise the denominator of the expression: $$\frac{1}{2 + \sqrt{3}}$$

Simplify: $$\sqrt{27} + \sqrt{12}$$

Simplify the following surd expression: $$ \sqrt{72} - \sqrt{18} $$

Simplify fully the expression: $$ \sqrt{125} + \sqrt{45} - \sqrt{80} $$

Express $$ \frac{15}{\sqrt{5}} $$ in its simplest surd form by rationalising the denominator.

(a) Express $$\frac{18}{\sqrt{3}}$$ in the form $$a\sqrt{b}$$, where $$a$$ and $$b$$ are integers.

(b) Given that $$x = \frac{12}{\sqrt{2}} - \sqrt{50}$$, find the value of $$x$$ in simplest surd form.

Given that $$( \sqrt{x} + \sqrt{y} )^2 = 27 + 12\sqrt{3}$$, where $$x$$ and $$y$$ are integers.

(a) Expand $$( \sqrt{x} + \sqrt{y} )^2$$.

(b) By comparing the expanded expression with $$27 + 12\sqrt{3}$$, find the values of $$x$$ and $$y$$. Assume $$x > y$$.

(c) Rationalise the denominator of $$\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}}$$ using the values of $$x$$ and $$y$$ found in part (b). Express your answer in simplest surd form.