Simplifying surds | Year 10

01 / LEARN

The idea, step by step

Separate a square factor

A square-root surd is exact. To simplify it, look for a square factor of the number under the root. For nonnegative $a, b$ , $\sqrt{a b} = \sqrt{a} \sqrt{b}$ .

Using the largest square factor usually finishes the calculation fastest. Since $72 = 36 \times 2$ , $\sqrt{72} = 6 \sqrt{2}$ .

Know when to stop

A simplest square-root surd has no square factor greater than $1$ left inside. For example, $\sqrt{12} = 2 \sqrt{3}$ , and $3$ has no further square factor.

With variables, $\sqrt{x^{2}} = ∣ x ∣$ . Only replace this by $x$ when $x \geq 0$ is stated. This prevents a sign error when the variable is negative.

A common trap

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

Instead: Square roots distribute over products of nonnegative numbers, not sums. $\sqrt{9 + 16} = 5$ , while $3 + 4 = 7$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A numerical surd

Simplify $\sqrt{180}$ .

$180 = 36 \times 5$ .
$\sqrt{180} = \sqrt{36} \sqrt{5} = 6 \sqrt{5}$ .

6 \sqrt{5}

Example 2

A coefficient outside the root

Simplify $3 \sqrt{48}$ .

$48 = 16 \times 3$ , so $\sqrt{48} = 4 \sqrt{3}$ .
Multiply the outside coefficients: $3 \times 4 \sqrt{3} = 12 \sqrt{3}$ .

12 \sqrt{3}

Check your understanding

Simplify $\sqrt{98}$ .

Show answer

$7 \sqrt{2}$

$98 = 49 \times 2$ and $\sqrt{49} = 7$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Simplify $\sqrt{20}$ .
2 marks
Simplify $\sqrt{75}$ .
2 marks
Simplify $2 \sqrt{45}$ .
2 marks
Simplify $\sqrt{200}$ .
2 marks
Simplify $\sqrt{48 x^{2}}$ for $x \geq 0$ .
3 marks
A square has area $50 {cm}^{2}$ . Find its exact side length.
3 marks

Read the worked solutions

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

Simplify $\sqrt{20}$ .
2 marks
$20 = 4 \times 5$ .
$2 \sqrt{5}$
Simplify $\sqrt{75}$ .
2 marks
$75 = 25 \times 3$ .
$5 \sqrt{3}$
Simplify $2 \sqrt{45}$ .
2 marks
$\sqrt{45} = 3 \sqrt{5}$ , then multiply by $2$ .
$6 \sqrt{5}$
Simplify $\sqrt{200}$ .
2 marks
$200 = 100 \times 2$ .
$10 \sqrt{2}$
Simplify $\sqrt{48 x^{2}}$ for $x \geq 0$ .
3 marks
$48 x^{2} = 16 x^{2} \times 3$ ; use $\sqrt{x^{2}} = x$ under the stated condition.
$4 x \sqrt{3}$
A square has area $50 {cm}^{2}$ . Find its exact side length.
3 marks
The side is $\sqrt{50} = \sqrt{25 \times 2} = 5 \sqrt{2}$ cm.
$5 \sqrt{2} cm$

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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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