Rationalising Denominators Flashcards

All 5 cards in this deck

What does it mean to rationalise the denominator of a fraction?

Rewrite the fraction so that the denominator is a rational number, with no surd in it.
e.g. $\frac{1}{\sqrt{2}}$ becomes $\frac{\sqrt{2}}{2}$

What are the steps to rationalise a denominator of the form $k/\sqrt{a}$?

e.g. $\frac{6}{\sqrt{3}}$

1. Multiply numerator and denominator by $\sqrt{3}$
2. Denominator becomes rational: $\frac{6\sqrt{3}}{3}$
3. Simplify: $2\sqrt{3}$

What is the conjugate of $a + \sqrt{b}$?

e.g. the conjugate of $3 - \sqrt{5}$

$a - \sqrt{b}$; the conjugate of $3 - \sqrt{5}$ is $3 + \sqrt{5}$.
Their product is rational: $9 - 5 = 4$.

What are the steps to rationalise a denominator of the form $r \pm s\sqrt{a}$?

e.g. $\frac{1+\sqrt{5}}{3-\sqrt{5}}$

1. Multiply numerator and denominator by the conjugate $3+\sqrt{5}$
2. Denominator: $9 - 5 = 4$; numerator: $3 + \sqrt{5} + 3\sqrt{5} + 5 = 8 + 4\sqrt{5}$
3. Simplify fully: $2 + \sqrt{5}$

True or false? Rationalising the denominator changes the value of the fraction.

False. It gives an equal (equivalent) fraction — only the form changes, e.g. $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.