Rationalising denominators | Year 10

01 / LEARN

The idea, step by step

Multiply by one

Rationalising changes the form of a fraction, not its value. Multiply numerator and denominator by the same nonzero expression.

For $1 / \sqrt{a}$ , with $a > 0$ , multiplying by $\sqrt{a} / \sqrt{a}$ gives $\sqrt{a} / a$ . The denominator is now rational.

Use the conjugate

For a denominator such as $3 + \sqrt{2}$ , multiplying by $3 - \sqrt{2}$ removes the radical because $(3 + \sqrt{2}) (3 - \sqrt{2}) = 9 - 2$ .

Write the whole numerator product before simplifying. A rationalised denominator is useful for exact comparison and later algebra, not an instruction to round the answer.

A common trap

Multiply only the denominator by the root.

Instead: That changes the value. The numerator and denominator must be multiplied by the same nonzero expression.

02 / WORKED EXAMPLES

See the thinking

Example 1

One root below

Rationalise $5 / \sqrt{3}$ .

Multiply top and bottom by $\sqrt{3}$ .
The denominator is $3$ , giving $5 \sqrt{3} / 3$ .

5 \sqrt{3} / 3

Example 2

A binomial below

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

Multiply top and bottom by $3 - \sqrt{5}$ .
The denominator is $9 - 5 = 4$ .
$2 (3 - \sqrt{5}) / 4 = (3 - \sqrt{5}) / 2$ .

(3 - \sqrt{5}) / 2

Check your understanding

Rationalise $1 / \sqrt{7}$ .

Show answer

$\sqrt{7} / 7$

Multiply by $\sqrt{7} / \sqrt{7}$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

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

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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 ↗

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