Rationalising denominators | Year 10

01 / LEARN

Understand the idea.

Multiply by one

Rationalising replaces a fraction by an equal fraction whose denominator is rational. Multiply its numerator and denominator by the same nonzero expression; that ratio equals $1$ , so the value is preserved.

For a single root, use the matching root: $\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}$ for $a > 0$ . Simplify common factors afterwards and keep the answer exact.

Use the conjugate

For a denominator containing two terms, try its conjugate, which reverses the sign between them. The product $(u + v) (u - v) = u^{2} - v^{2}$ removes a square root when its square is rational.

Multiply the entire numerator as well as the entire denominator. For $3 + \sqrt{2}$ , the conjugate product is $9 - 2 = 7$ . Check that the chosen multiplier is nonzero before treating its ratio as $1$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

One root below

Rationalise $5 / \sqrt{3}$ .

Worked solution

1
Choose a value-preserving multiplier.
$\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$
The multiplier equals $1$ because $\sqrt{3} \neq 0$ .
2
Multiply numerator and denominator.
$\frac{5 \sqrt{3}}{\sqrt{3} \sqrt{3}} = \frac{5 \sqrt{3}}{3}$
The two denominator roots multiply to $3$ .
3
Check the exact form.
$\frac{5}{\sqrt{3}} = \frac{5 \sqrt{3}}{3}$
The denominator is rational and there is no common integer factor to cancel.

Answer

$5 \sqrt{3} / 3$

EXAMPLE 02

A binomial below

Rationalise $2 / (3 + \sqrt{5})$ .

Worked solution

1
Use the conjugate.
$\frac{2}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}}$
The conjugate is nonzero and its product with the denominator is a difference of squares.
2
Evaluate the denominator product.
$\frac{2 (3 - \sqrt{5})}{(3 + \sqrt{5}) (3 - \sqrt{5})} = \frac{2 (3 - \sqrt{5})}{9 - 5}$
The opposite middle terms cancel.
3
Cancel the common factor.
$\frac{2 (3 - \sqrt{5})}{4} = \frac{3 - \sqrt{5}}{2}$
Both numerator and denominator contain a factor of $2$ .

Answer

$(3 - \sqrt{5}) / 2$

03 / YOUR TURN

Try it yourself.

Check your understanding

Rationalise $1 / \sqrt{7}$ .

Show answer & reasoning

$\sqrt{7} / 7$

Multiply by $\frac{\sqrt{7}}{\sqrt{7}}$ , which equals $1$ : $\frac{1}{\sqrt{7}} = \frac{\sqrt{7}}{7}$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

Rationalise $\frac{3}{\sqrt{2}}$ .
2 marks
Rationalise $\frac{4}{\sqrt{8}}$ .
2 marks
Rationalise $\frac{1}{2 + \sqrt{3}}$ .
3 marks
Rationalise $\frac{3}{\sqrt{5} - 1}$ .
3 marks
Simplify and rationalise $\frac{\sqrt{6}}{\sqrt{2}}$ .
2 marks
A rectangle has area $12 {cm}^{2}$ and width $\sqrt{3}$ cm. Find its exact length.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Rationalise $\frac{3}{\sqrt{2}}$ .
2 marks
Multiply numerator and denominator by $\sqrt{2}$ .
$\frac{3 \sqrt{2}}{2}$
Rationalise $\frac{4}{\sqrt{8}}$ .
2 marks
$\frac{4}{2 \sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$ .
$\sqrt{2}$
Rationalise $\frac{1}{2 + \sqrt{3}}$ .
3 marks
The conjugate product is $4 - 3 = 1$ .
$2 - \sqrt{3}$
Rationalise $\frac{3}{\sqrt{5} - 1}$ .
3 marks
Multiply by $\frac{\sqrt{5} + 1}{\sqrt{5} + 1}$ ; the denominator becomes $5 - 1 = 4$ .
$\frac{3 (\sqrt{5} + 1)}{4}$
Simplify and rationalise $\frac{\sqrt{6}}{\sqrt{2}}$ .
2 marks
Combine the quotient as $\sqrt{\frac{6}{2}} = \sqrt{3}$ .
$\sqrt{3}$
A rectangle has area $12 {cm}^{2}$ and width $\sqrt{3}$ cm. Find its exact length.
3 marks
Length $= \frac{12}{\sqrt{3}} = \frac{12 \sqrt{3}}{3} = 4 \sqrt{3}$ cm.
$4 \sqrt{3} cm$

Checking saved progress…

Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

In this lesson

Understand the idea.

Multiply by one

Use the conjugate

See the method in action.

One root below

Choose a value-preserving multiplier.

Multiply numerator and denominator.

Check the exact form.

A binomial below

Use the conjugate.

Evaluate the denominator product.

Cancel the common factor.

Try it yourself.

Check your understanding

Keep practising