Rationalising the denominator

Rationalising the denominator

When we have a surd in the denominator of a fraction, we can rationalise the denominator by multiplying both the numerator and denominator by the surd (or the negated factor).

Example: rationalise \frac{1}{\sqrt{2}}

• =\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}
• =\frac{\sqrt{2}}{2}

Example: rationalise \frac{3}{\sqrt{5}}

• =\frac{3\times\sqrt{5}}{\sqrt{5}\times\sqrt{5}}
• =\frac{3\sqrt{5}}{5}

Example: rationalise \frac{4}{\sqrt{3}}

• =\frac{4\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}
• =\frac{4\sqrt{3}}{3}

Example: rationalise \frac{\sqrt{2}}{1+\sqrt{3}}

• =\frac{\sqrt{2}(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}
• =\frac{\sqrt{2}-\sqrt{6}}{1-3}
• =\frac{\sqrt{2}-\sqrt{6}}{-2}

Example: rationalise \frac{5}{2+\sqrt{7}}

• =\frac{5(2-\sqrt{7})}{(2+\sqrt{7})(2-\sqrt{7})}
• =\frac{10-5\sqrt{7}}{4-7}
• =\frac{10-5\sqrt{7}}{-3}

Example: rationalise \frac{3\sqrt{5}}{4-\sqrt{2}}

• =\frac{3\sqrt{5}(4+\sqrt{2})}{(4-\sqrt{2})(4+\sqrt{2})}
• =\frac{12\sqrt{5}+3\sqrt{10}}{16-2}
• =\frac{12\sqrt{5}+3\sqrt{10}}{14}