Rationalising surds

In a nutshell

Single or double brackets involving surds can be expanded or multiplied out in a similar way to expanding single or double brackets with algebraic expressions. Rationalising surds eliminates the irrational number in the denominator. You can think of this as converting to equivalent fractions with a rational denominator. Surd fractions with more complex denominators will need the conjugate surd to rationalise them.

Definitions

Rationalising	Converting a fraction with an irrational denominator to one with a rational denominator.
Conjugate	A binomial expression with the same terms as the original, with the opposite sign in the middle.

Expanding double brackets with surds

To expand double brackets, you can treat surds as you would algebraic terms. Make sure to multiply each term in the first bracket by each term in the second bracket. You can use the F.O.I.L. method or a multiplication grid.

FOIL

First	Multiply the first terms of each bracket.
Outside	Multiply the outside terms.
Inside	Multiply the two inside terms.
Last	Multiply the last terms of each bracket.

Note: Once the brackets have been multiplied out, simplify the terms where possible.

Maths; Number; KS4 Year 10; Rationalising surds - Higher

Example 1

Expand $(1+\sqrt2)(2+\sqrt{3})$ using the F.O.I.L. method.

$\begin{aligned}(1+\sqrt2)(2+\sqrt{3})&= \underbrace{2}_{\text{First}}+\underbrace{\sqrt{3}}_{\text{Outside}}+\underbrace{2\sqrt2}_{\text{Inside}}+\underbrace{\sqrt6}_{\text{Last}} \\&=\underline{2+\sqrt3+2\sqrt2+\sqrt6}\end{aligned}$

MULTIPLICATION GRID

A multiplication grid separates each term of either bracket as headers for rows and columns. The rest of the table is filled with the corresponding products.

Example 2

Expand $(4+\sqrt3)(1-\sqrt3)$ .

Build your multiplication grid.

$\times$	$4$	$\sqrt3$
$1$	$4$	$\sqrt3$
$-\sqrt3$	$-4\sqrt3$	$-3$

Add the terms and simplify.

$\begin {aligned} (4+\sqrt3)(1-\sqrt3) &= 4 + \sqrt3 -4\sqrt3 -3 \\ &= \underline{1-3\sqrt3} \end {aligned}$

Example 3

Multiply out and simplify $(4+\sqrt3)(1+\sqrt3)$ .

$\begin {aligned} (4+\sqrt3)(1+\sqrt3) &= 4 + 4\sqrt3 +\sqrt3 +3 \\ &= \underline{7+5\sqrt3} \end {aligned}$ 

Rationalising the denominator

Rationalising the denominator means removing the surd from the bottom of a fraction, by converting to an equivalent fraction. This is done by multiplying the numerator and the denominator by the same term.

To rationalise a surd, you multiply the numerator and the denominator by the surd in the denominator.

Example 4

Rationalise the denominator $\dfrac{1}{\sqrt5}$ .

Multiply both the numerator and the denominator by the surd in the denominator. Here, it is $\sqrt5$ . So:

$\begin {aligned} \frac{1}{\sqrt5} &= \frac{1\times \sqrt5}{\sqrt5 \times \sqrt5}\\ &= \underline{\frac{\sqrt5}{5}} \end {aligned}$ 

Rationalising for more complex denominators

Sometimes you can have more complex denominators like:

$\dfrac{1}{2+\sqrt3}$ 

Multiplying top and bottom by $\sqrt3$ will still leave a surd in the denominator. You need the conjugate expression of the denominator to eliminate the surd. In this case, the conjugate is:

$2-\sqrt3$

procedure

1.	Form the conjugate of the denominator.
2.	Multiply both the numerator and the denominator by the conjugate.
3.	Simplify the answer.

Example 5

Rationalise the denominator $\frac{10}{3+\sqrt2}$ .

The conjugate is $3-\sqrt2$ .

Multiply both the numerator and the denominator by the conjugate.

$\begin {aligned}\frac{10}{3+\sqrt2} &= \frac{10\times (3-\sqrt2)}{(3+\sqrt2)(3-\sqrt2)}\\\\ &=\frac{30-10\sqrt2}{9-3\sqrt2+3\sqrt3-2}\\\\&= \underline{\frac{30-10\sqrt2}{7}} \end {aligned}$ 