Main problem

Each line is a set of equivalent fractions. Fill in the blanks in the fractions to make each line complete, including the multiplier used to get from one fraction to the next.

  1. \(\dfrac{1}{\sqrt{2}} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{\sqrt{2}}{\quad\quad\quad} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{\sqrt{6}}{\quad\quad\quad} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{\quad\quad\quad}{6}\)

  1. \(\dfrac{2}{5\sqrt{3}} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{\quad\quad\quad}{15} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{2\sqrt{6}}{\quad\quad\quad} \, \bigg(\times \dfrac{\quad}{\quad\quad}\bigg) = \dfrac{\quad\quad\quad}{60}\)

  1. \(\dfrac{5}{2+\sqrt{2}} \, \bigg(\times \dfrac{\quad}{\quad\quad\quad}\bigg) = \dfrac{10-5\sqrt{2}}{\quad\quad\quad} \, \bigg(\times \dfrac{\quad}{\quad\quad\quad}\bigg) = \dfrac{\quad\quad\quad}{20+10\sqrt{2}}\)

  1. \(\dfrac{2-\sqrt{3}}{4} \, \bigg(\times \dfrac{\quad}{\quad\quad\quad}\bigg) = \dfrac{\quad\quad\quad\quad}{8+4\sqrt{3}} \, \bigg(\times \dfrac{\quad}{\quad\quad\quad}\bigg) = \dfrac{\quad\quad\quad\quad}{16}\)

A rationalised fraction is one whose denominator is a whole number. These are usually easier to work with than fractions with square roots in their denominators.

  • Identify the rationalised fractions in the above lines. What do you notice about the multipliers when moving from a fraction with a surd (square root) in the denominator to a rationalised fraction?

  • How would you rationalise fractions in the following form: \(\dfrac{a}{\sqrt{b}}\), \(\dfrac{a}{b\sqrt{c}}\) and \(\dfrac{a}{b+\sqrt{c}}\)?

  • Is there more than one way to rationalise a fraction?