Rationalising Denominators

Rationalising Denominatorsvideo

Rationalising denominators

What does rationalising the denominator mean?

• If a fraction has a denominator containing a surd then it has an irrational denominator

  • E.g. $\frac{4}{\sqrt{5}}$ or $\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}}$

• The fraction can be rewritten as an equivalent fraction, but with a rational denominator

  • E.g. $\frac{4\sqrt{5}}{5}$ or $\frac{\sqrt{6}}{3}$

• The numerator may contain a surd, but the denominator is rationalised

How do I rationalise simple denominators?

• If the denominator is a surd:

  • Multiply the top and bottom of the fraction by the surd on the denominator

    • $\frac{a}{\sqrt{b}}= \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}}$

    • This is equivalent to multiplying by 1, so does not change the value of the fraction

    • $\sqrt{b} \times \sqrt{b} = b$ so the denominator is no longer a surd

  • Multiply the fractions as you would usually, and simplify if needed

    • $\frac{a\sqrt{b}}{b}$

How do I rationalise harder denominators?

• If the denominator is an expression containing a surd:

  • For example $\frac{2}{3 + \sqrt{5}}$ 

  • Multiply the top and bottom of the fraction by the expression on the denominator, but with the sign changed

    • $\frac{2}{3+\sqrt{5}}=\frac{2}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$

    • This is equivalent to multiplying by 1, so does not change the value of the fraction

  • Multiply the fractions as you would usually (use brackets to help)

    • $\frac{2}{3+\sqrt{5}}=\frac{2\left(3-\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}$

  • Expand any brackets, and simplify

    • $\frac{2}{3+\sqrt{5}}=\frac{6-2\sqrt{5}}{\left(3\times 3\right)+3\sqrt{5}-3\sqrt{5}-\left(\sqrt{5}\sqrt{5}\right)}=\frac{6-2\sqrt{5}}{9-\left(5\right)}$

  • You can use the difference of two squares to expand the denominator quickly

    • $\left(a+\sqrt{b}\right)\left(a-\sqrt{b}\right)=a^2-{\left(\sqrt{b}\right)}^2=a^2-b$

    • This is what makes the denominator rational

  • Simplify

    • $\frac{2}{3+\sqrt{5}}=\frac{6-2\sqrt{5}}{4}=\frac{3-\sqrt{5}}{2}$