YEAR 10 · EXPONENTIAL RELATIONSHIPS · 15 MIN

Simplifying surds

Extract square factors to write surds in their simplest exact form.

In this lesson

Identify a square factor
Simplify a surd without replacing it by a decimal

Before you start

Factor pairs
Square roots

01 / LEARN

Understand the idea.

Separate a square factor

To simplify a square-root surd, factor its radicand into a perfect square and what remains. For nonnegative $a, b$ , $\sqrt{a b} = \sqrt{a} \sqrt{b}$ , so a square factor can leave the root as its square root.

Look for the largest square factor rather than any factor. In $\sqrt{72}$ , using $72 = 36 \times 2$ gives $6 \sqrt{2}$ immediately. Starting with $4 \times 18$ also works, but needs another simplification.

Know when to stop

A numerical square-root surd is fully simplified when its remaining positive integer radicand has no square factor greater than $1$ . Keep this exact form unless the question requests a decimal approximation.

Variable factors need a sign check: $\sqrt{x^{2}} = ∣ x ∣$ because a square root is nonnegative. If $x \geq 0$ is stated, this becomes $x$ ; if $x < 0$ , it becomes $- x$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A numerical surd

Simplify $\sqrt{180}$ .

Worked solution

1
Find a square factor.
$180 = 36 \times 5$
The factor $36$ is the largest perfect square dividing $180$ .
2
Separate the roots.
$\sqrt{180} = \sqrt{36} \sqrt{5}$
Both factors are nonnegative, so the product rule for square roots applies.
3
Extract the perfect square.
$\sqrt{180} = 6 \sqrt{5}$
Since $\sqrt{36} = 6$ and $5$ has no square factor greater than $1$ , the surd is simplified.

Answer

$6 \sqrt{5}$

EXAMPLE 02

A coefficient outside the root

Simplify $3 \sqrt{48}$ .

Worked solution

1
Factor the radicand.
$48 = 16 \times 3$
Extracting the largest square factor will leave a simplified radical.
2
Simplify the root.
$\sqrt{48} = \sqrt{16} \sqrt{3} = 4 \sqrt{3}$
The perfect-square factor contributes an outside coefficient of $4$ .
3
Multiply the outside coefficients.
$3 \sqrt{48} = 3 (4 \sqrt{3}) = 12 \sqrt{3}$
The original coefficient multiplies the extracted coefficient.

Answer

$12 \sqrt{3}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $\sqrt{98}$ .

Show answer & reasoning

$7 \sqrt{2}$

Use the square factor $49$ : $\sqrt{98} = \sqrt{49 \times 2} = 7 \sqrt{2}$ . The remaining radicand $2$ has no further square factor.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Separate a square factor

Know when to stop

See the method in action.

A numerical surd

Find a square factor.

Separate the roots.

Extract the perfect square.

A coefficient outside the root

Factor the radicand.

Simplify the root.

Multiply the outside coefficients.

Try it yourself.

Check your understanding

Keep practising