Rational/irrational numbers and surd operations | Year 11

01 / LEARN

The idea, step by step

Which roots are irrational?

A rational number is a ratio of integers with non-zero denominator. Its decimal terminates or repeats. An irrational decimal neither terminates nor repeats; $\sqrt{2}$ is an example.

A root is not automatically irrational: $\sqrt{49} = 7$ . Simplify a radical by extracting square factors. For instance, $\sqrt{20} = \sqrt{4 \times 5} = 2 \sqrt{5}$ .

Collect like radicals and rationalise

After simplifying, add coefficients of the same radical just as you collect like algebraic terms. Products of square roots can combine when their radicands are non-negative.

To remove a radical denominator, multiply numerator and denominator by a suitable radical or conjugate. This is multiplication by 1. Keep exact forms until the final requested approximation.

A common trap

Using $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ .

Instead: Roots distribute over non-negative products, not over sums.

02 / WORKED EXAMPLES

See the thinking

Example 1

Simplify then collect

Simplify $\sqrt{72} - 2 \sqrt{8}$ .

$\sqrt{72} = 6 \sqrt{2}$ and $\sqrt{8} = 2 \sqrt{2}$ .
$6 \sqrt{2} - 4 \sqrt{2} = 2 \sqrt{2}$ .

2 \sqrt{2}

Example 2

Use a conjugate

Rationalise $3 / (\sqrt{5} - 1)$ .

Multiply numerator and denominator by $\sqrt{5} + 1$ .
The denominator is $5 - 1 = 4$ .

3 (\sqrt{5} + 1) / 4

Check your understanding

Is $\sqrt{0.04}$ rational?

Show answer

Yes: 0.2.

${0.2}^{2} = 0.04$ and $0.2 = 1 / 5$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Classify $\sqrt{81}$ and $\sqrt{18}$ .
2 marks
Simplify $\sqrt{50}$ .
2 marks
Simplify $3 \sqrt{12} - \sqrt{27}$ .
2 marks
Expand $(\sqrt{7} + 2) (\sqrt{7} - 2)$ .
2 marks
Rationalise $5 / \sqrt{3}$ .
2 marks
Rationalise $2 / (\sqrt{6} - 2)$ .
3 marks

Read the worked solutions

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

Classify $\sqrt{81}$ and $\sqrt{18}$ .
2 marks
$\sqrt{81} = 9$ ; $\sqrt{18} = 3 \sqrt{2}$ .
9 is rational; the square root of 18 is irrational.
Simplify $\sqrt{50}$ .
2 marks
$50 = 25 \times 2$ so extract $\sqrt{25} = 5$ .
$5 \sqrt{2}$
Simplify $3 \sqrt{12} - \sqrt{27}$ .
2 marks
$3 (2 \sqrt{3}) - 3 \sqrt{3} = 3 \sqrt{3}$ .
$3 \sqrt{3}$
Expand $(\sqrt{7} + 2) (\sqrt{7} - 2)$ .
2 marks
Use difference of squares: $7 - 4 = 3$ .
3
Rationalise $5 / \sqrt{3}$ .
2 marks
Multiply top and bottom by $\sqrt{3}$ .
$5 \sqrt{3} / 3$
Rationalise $2 / (\sqrt{6} - 2)$ .
3 marks
Multiply by the conjugate: $2 (\sqrt{6} + 2) / (6 - 4)$ .
Cancel the factor 2.
$\sqrt{6} + 2$

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

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