Operations with surds | Year 10

01 / LEARN

The idea, step by step

Like surds behave like like terms

Simplify each radical before combining. Terms with the same root part can be added or subtracted by combining their coefficients.

For example, $2 \sqrt{3} + 5 \sqrt{3} = 7 \sqrt{3}$ , while $2 \sqrt{3} + 5 \sqrt{2}$ cannot combine into a single like term.

Products use distribution

For nonnegative radicands, multiply square roots by multiplying the radicands. Expand brackets with the distributive law just as in algebra, then simplify.

Conjugate pairs are especially useful: $(a + b) (a - b) = a^{2} - b^{2}$ . If $a$ or $b$ is a square root, the squared term may become rational.

A common trap

$\sqrt{2} + \sqrt{3} = \sqrt{5}$ .

Instead: Addition does not combine different roots. The expression is already exact and cannot be simplified this way.

02 / WORKED EXAMPLES

See the thinking

Example 1

Simplify before adding

Simplify $\sqrt{12} + \sqrt{27}$ .

$\sqrt{12} = 2 \sqrt{3}$ and $\sqrt{27} = 3 \sqrt{3}$ .
Add like terms: $2 \sqrt{3} + 3 \sqrt{3} = 5 \sqrt{3}$ .

5 \sqrt{3}

Example 2

Expand a product

Expand $(\sqrt{5} + 2) (\sqrt{5} - 2)$ .

The factors are conjugates, so use difference of squares.
${(\sqrt{5})}^{2} - 2^{2} = 5 - 4 = 1$ .

Check your understanding

Simplify $\sqrt{6} \sqrt{15}$ .

Show answer

$3 \sqrt{10}$

The product is $\sqrt{90} = 3 \sqrt{10}$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $4 \sqrt{2} + 3 \sqrt{2}$ .
2 marks
Simplify $\sqrt{18} - \sqrt{8}$ .
2 marks
Find $\sqrt{3} \sqrt{12}$ .
2 marks
Expand ${(2 + \sqrt{3})}^{2}$ .
3 marks
Simplify $\sqrt{75} / \sqrt{3}$ .
2 marks
A rectangle has sides $2 \sqrt{3}$ cm and $3 \sqrt{6}$ cm. Find its exact area.
3 marks

Read the worked solutions

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

Simplify $4 \sqrt{2} + 3 \sqrt{2}$ .
2 marks
Add coefficients: $4 + 3 = 7$ .
$7 \sqrt{2}$
Simplify $\sqrt{18} - \sqrt{8}$ .
2 marks
$3 \sqrt{2} - 2 \sqrt{2} = \sqrt{2}$ .
$\sqrt{2}$
Find $\sqrt{3} \sqrt{12}$ .
2 marks
$\sqrt{36} = 6$ .
6
Expand ${(2 + \sqrt{3})}^{2}$ .
3 marks
$4 + 4 \sqrt{3} + 3 = 7 + 4 \sqrt{3}$ .
$7 + 4 \sqrt{3}$
Simplify $\sqrt{75} / \sqrt{3}$ .
2 marks
$\sqrt{75 / 3} = \sqrt{25} = 5$ .
5
A rectangle has sides $2 \sqrt{3}$ cm and $3 \sqrt{6}$ cm. Find its exact area.
3 marks
$2 \sqrt{3} \times 3 \sqrt{6} = 6 \sqrt{18} = 18 \sqrt{2}$ .
$18 \sqrt{2} {cm}^{2}$

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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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