Chapter 3 · Additional Maths · A3 Surds

Surds

A surd is an exact square root such as $\sqrt{2}$ . Its decimal never ends and never repeats, so we keep it as a root instead of rounding it. By the end of this chapter you can simplify surds, expand and rationalise with them, find unknown constants by comparing parts, solve equations that contain them, and give exact answers in geometry.

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The laws

Roots split over multiplication and division, never over addition. Take out the largest perfect square to reach simplest form.

The conjugate

To rationalise $p + q\sqrt{r}$ multiply top and bottom by $p - q\sqrt{r}$ . The bottom becomes the whole number $p^2 - q^2 r$ .

Test every root

Isolate the root, square the whole side, solve, then test each answer in the original. A negative right-hand side is a rejected answer.

By the end of this chapter

Tell a surd from a rational number, and place a surd between two whole numbers.
Use the laws of surds to simplify roots, then collect like surds into simplest form.
Expand brackets that contain surds, including the squares and the difference of two squares.
Rationalise a denominator with one term or with two terms using the conjugate, and find constants by comparing rational and irrational parts.
Solve equations with surds in the coefficients or with the unknown under a root, reject answers that fail the test, and give exact lengths and areas.

✶

Start here

Maths In Real Life

Surds are the topic of exact answers, so one real situation is enough to show why they matter: lengths that come from squares and diagonals are rarely tidy decimals. Pure topics such as this one have fewer real-life uses than curves do, so there is one problem here with four parts. The picture comes first, then the maths is drawn on top by script.

$A square rooftop garden at an HDB block seen from above, with its boundary outlined, a dashed diagonal path d, the side s and the area of 50 square metres marked over the picture$

d

s

50\text{ m}^2

Maths In Real Life 1

The rooftop garden

Planners of rooftop gardens often lay a square plot and run a path along its diagonal. A square's diagonal is always its side times $\sqrt{2}$ , so the path is a surd unless the side is itself a multiple of $\sqrt{2}$ .

A rooftop garden at an HDB block is a square plot of area $50$ square metres. Let the side of the plot be $s$ metres. A straight path of length $d$ metres runs from one corner to the opposite corner.

(a)Find the length of one side of the plot in the form $a\sqrt{b}$ metres.
(b)Find the exact length of the path.
(c)A fence goes round the whole plot. Find its length in the form $a\sqrt{b}$ metres.
(d)Fencing costs $12 per metre. Find the exact cost of the fence, and its cost to the nearest cent.

Show full solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

(a)

\displaystyle \sqrt{50}\;= \sqrt{\textcolor{#010104}{25} \times 2} = \textcolor{#010106}{5\sqrt{2}}

50

has the perfect-square factor

25

(b)

The diagonal splits the square into two right-angled triangles whose other sides are both the side of the square.

\displaystyle (5\sqrt{2})^2 + (5\sqrt{2})^2\;= 50 + 50 = 100

Pythagoras: each square of a side is the area.

\displaystyle \sqrt{100}\;= \textcolor{#010106}{10}

Take the root and simplify.

(c)

\displaystyle 4 \times 5\sqrt{2}\;= \textcolor{#010106}{20\sqrt{2}}

A square has four equal sides.

(d)

\displaystyle 12 \times 20\sqrt{2}\;= \textcolor{#010106}{240\sqrt{2}}

Cost in dollars = price per metre times length in metres. Keep the surd exact.

For a price in dollars, use a calculator:

240\sqrt{2} = 339.41

to 2 decimal places, so the cost is about $339.41.

Round only at the very end.

Answer

(a)

5\sqrt{2}\text{ m}

(b)

10\text{ m}

(c)

20\sqrt{2}\text{ m}

(d) $

240\sqrt{2}

exactly, which is $339.41 to the nearest cent

Supports K341-A3-01

What is a surd? Rational and irrational numbers

Revision. A rational number can be written as a fraction of two whole numbers, such as

\dfrac{3}{4}

-5

0.2

\sqrt{49}

(which is just

7

). Its decimal either stops or repeats.

Take $\sqrt{2}$ . No fraction multiplied by itself gives exactly $2$ , and its decimal $1.41421\ldots$ never stops and never repeats. A number like this is irrational. When an irrational number is the root of a rational number, it is called a surd.

The Core Idea

A surd is an irrational root, so we leave it as a root.

$\sqrt{n}$ is a surd exactly when $n$ is not the square of a rational number. $\sqrt{12}$ is a surd. $\sqrt{16}$ is not, because it is just $4$ . All the surds in this chapter are square roots.

Case table 1: a root is a surd only when the number under it is not a perfect square
Number	Simplest form	Rational or surd?
$\sqrt{16}$	$4$	rational (not a surd)
$\sqrt{20}$	$2\sqrt{5}$	surd
$\sqrt{36}$	$6$	rational (not a surd)
$\sqrt{50}$	$5\sqrt{2}$	surd
$\sqrt{100}$	$10$	rational (not a surd)

The diagonal of a unit square has length

\sqrt{2}

. A compass arc carries it onto the number line.

Surds sit between whole numbers:

\sqrt{2}

\sqrt{5}

and

\sqrt{8}

(which is

2\sqrt{2}

) on one line.

SGSchoolKaki Tip

A calculator shows $\sqrt{2}$ as $1.414213562$ , but that is a rounded copy. In this chapter the exact surd is the answer. Decimals are for checking only.

Think About It

Can two surds multiply to give a rational number? Can they add to give one? Try

\sqrt{2} \times \sqrt{8}

first.

Show the full workingHide the working

Colour keyfinal form or accepted answer

\displaystyle \sqrt{2} \times \sqrt{8}\;= \sqrt{16} = \textcolor{#010106}{4}

Multiply under one root.

The product is rational, even though both numbers are irrational.

\displaystyle (3 + \sqrt{2}) + (3 - \sqrt{2})\;= \textcolor{#010106}{6}

The surd parts cancel.

The sum is rational too: the two surds were opposites.

Answer

Yes to both. A product or a sum of surds can be rational, so never assume that a calculation with surds stays irrational.

Example 1.1

Surd or rational?

Decide which of these numbers are surds, and give a reason for each: $\sqrt{12}$ , $\sqrt{49}$ , $\sqrt{0.16}$ , $\sqrt{20}$ , $\sqrt{7}$ .

Show solutionHide solution

\sqrt{12}

: there is no whole number or fraction that squares to

12

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

\sqrt{49}

: the number under the root is a perfect square, so

\sqrt{49} = 7

. It is rational, not a surd.

Rational means it can be written as a fraction.

\sqrt{0.16}

: the number under the root is the square of a fraction, so

\sqrt{0.16} = 0.4

. It is rational, not a surd.

Rational means it can be written as a fraction.

\sqrt{20}

: there is no whole number or fraction that squares to

20

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

\sqrt{7}

: there is no whole number or fraction that squares to

7

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

Answer

Surds:

\sqrt{12}

\sqrt{20}

\sqrt{7}

. Rational:

\sqrt{49}

\sqrt{0.16}

Your Turn 1.1

Decide which of these numbers are surds, and give a reason for each: $\sqrt{18}$ , $\sqrt{64}$ , $\sqrt{2.25}$ , $\sqrt{15}$ , $\sqrt{50}$ .

Show full solutionHide solution

\sqrt{18}

: there is no whole number or fraction that squares to

18

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

\sqrt{64}

: the number under the root is a perfect square, so

\sqrt{64} = 8

. It is rational, not a surd.

Rational means it can be written as a fraction.

\sqrt{2.25}

: the number under the root is the square of a fraction, so

\sqrt{2.25} = 1.5

. It is rational, not a surd.

Rational means it can be written as a fraction.

\sqrt{15}

: there is no whole number or fraction that squares to

15

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

\sqrt{50}

: there is no whole number or fraction that squares to

50

, so its decimal never ends and never repeats. It is irrational, so it is a surd.

An irrational root of a rational number.

Answer

Surds:

\sqrt{18}

\sqrt{15}

\sqrt{50}

. Rational:

\sqrt{64}

\sqrt{2.25}

Example 1.2

Between which whole numbers?

Between which two consecutive whole numbers does each of these lie? $\sqrt{10}$ , $\sqrt{40}$ , $\sqrt{90}$ .

Show solutionHide solution

3^2 = 9

and

4^2 = 16

, and

9 < 10 < 16

, so

3 < \sqrt{10} < 4

Find the two perfect squares either side.

6^2 = 36

and

7^2 = 49

, and

36 < 40 < 49

, so

6 < \sqrt{40} < 7

Find the two perfect squares either side.

9^2 = 81

and

10^2 = 100

, and

81 < 90 < 100

, so

9 < \sqrt{90} < 10

Find the two perfect squares either side.

Answer

3 < \sqrt{10} < 4

;

6 < \sqrt{40} < 7

;

9 < \sqrt{90} < 10

Example 1.3

Rational or irrational?

Say whether each of these numbers is rational or irrational, and show how you know: $\sqrt{2} \times \sqrt{8}$ ; $2 + \sqrt{5}$ ; $(\sqrt{7})^2$ ; $\sqrt{2} \times \sqrt{3}$ .

Show solutionHide solution

\displaystyle \sqrt{2} \times \sqrt{8}\;= \sqrt{16} = 4

Multiply under one root, then simplify.

4

is a rational number.

2 + \sqrt{5}

is already in simplest form: a whole number plus a surd.

2 + \sqrt{5}

is irrational: a rational number plus a surd is never rational (if it were, taking away the rational part would make the surd rational).

\displaystyle (\sqrt{7})^2\;= 7

A root multiplied by itself gives the number underneath.

7

is a rational number.

\displaystyle \sqrt{2} \times \sqrt{3}\;= \sqrt{6}

Multiply under one root, then simplify.

\sqrt{6}

is irrational: it is a non-zero multiple of the surd

\sqrt{6}

Answer

4

: rational;

2 + \sqrt{5}

: irrational;

7

: rational;

\sqrt{6}

: irrational

Your Turn 1.3

Say whether each of these numbers is rational or irrational, and show how you know: $\sqrt{3} \times \sqrt{12}$ ; $3 + \sqrt{7}$ ; $(\sqrt{11})^2$ ; $\sqrt{2} \times \sqrt{5}$ .

Show full solutionHide solution

\displaystyle \sqrt{3} \times \sqrt{12}\;= \sqrt{36} = 6

Multiply under one root, then simplify.

6

is a rational number.

3 + \sqrt{7}

is already in simplest form: a whole number plus a surd.

3 + \sqrt{7}

is irrational: a rational number plus a surd is never rational (if it were, taking away the rational part would make the surd rational).

\displaystyle (\sqrt{11})^2\;= 11

A root multiplied by itself gives the number underneath.

11

is a rational number.

\displaystyle \sqrt{2} \times \sqrt{5}\;= \sqrt{10}

Multiply under one root, then simplify.

\sqrt{10}

is irrational: it is a non-zero multiple of the surd

\sqrt{10}

Answer

6

: rational;

3 + \sqrt{7}

: irrational;

11

: rational;

\sqrt{10}

: irrational

Supports K341-A3-01

The laws of surds and simplest form

Surds follow two laws that let you tidy any root. A root of a product splits into a product of roots, and a root of a quotient splits into a quotient of roots. Together with $(\sqrt{a})^2 = a$ they are all you need to simplify.

The Core Idea

Roots split over multiplication and division, never over addition.

For a non-negative $a$ and $b$ we have $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ , and $\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$ when $b$ is positive. Reading the first law backwards is how you simplify.

Case table 2: the laws of surds, with numbers
Law	In symbols	With numbers
Product	$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$	$\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$
Quotient	$\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}$	$\dfrac{\sqrt{50}}{\sqrt{2}} = \sqrt{25} = 5$
Square	$(\sqrt{a})^2 = a$	$(\sqrt{7})^2 = 7$
Not allowed	$\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}$	$\sqrt{9 + 16} = 5$ but $\sqrt{9} + \sqrt{16} = 7$

Watch out.

\sqrt{9 + 16}

\sqrt{25} = 5

, not

\sqrt{9} + \sqrt{16} = 7

. Never split a root over a sum or a difference.

Simplest form and like surds

A surd is in simplest form when the number under the root has no perfect-square factor except $1$ . To get there, take out the largest perfect square factor ( $4$ , $9$ , $16$ , $25$ , $36$ , and so on). Surds with the same number under the root are like surds, and only like surds can be added or subtracted, exactly as $2x + 5x$ gives $7x$ .

The Core Idea

Simplify every surd first, then collect the like surds.

Write each number under a root as a perfect square times something smaller, take the root of the perfect square out, then add or subtract the like surds.

SGSchoolKaki Tip

If you pull out a smaller square than the largest, nothing is lost: $\sqrt{72}$ becomes $2\sqrt{18}$ , and you simply keep going until no square factor is left.

Think About It

A student says that

\sqrt{9 + 16}

is the same as

\sqrt{9} + \sqrt{16}

, which is

7

. Find the mistake.

Show the full workingHide the working

Colour keyfinal form or accepted answer

\displaystyle \sqrt{9 + 16}\;= \sqrt{25} = \textcolor{#010106}{5}

Add first, then take the root.

\displaystyle \sqrt{9} + \sqrt{16}\;= 3 + 4 = 7

The student's total.

The two totals are different, so splitting

\sqrt{9 + 16}

over the plus sign is false.

Answer

A root cannot be split over an addition. Only products and quotients split: the correct value of

\sqrt{9 + 16}

5

Links To

Roots of quadratics often arrive as surds, for example from the quadratic formula in Chapter 1: Quadratic functions

Example 2.1

Take out a perfect square

Write $\sqrt{72}$ in the form $a\sqrt{b}$ , where $a$ and $b$ are integers and $b$ is as small as possible.

Show solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle \sqrt{72}\;= \sqrt{\textcolor{#010104}{36} \times 2}

Look for the largest perfect square that divides

72

. It is

36

, because

72 = 36 \times 2

\displaystyle \phantom{\sqrt{72}}\;= \textcolor{#010104}{\sqrt{36}} \times \sqrt{2}

Split the root:

\sqrt{xy} = \sqrt{x} \times \sqrt{y}

\displaystyle \phantom{\sqrt{72}}\;= \textcolor{#010106}{6\sqrt{2}}

The root of the perfect square comes out:

\sqrt{36} = 6

Answer

6\sqrt{2}

Your Turn 2.1

Write $\sqrt{75}$ in the form $a\sqrt{b}$ , where $a$ and $b$ are integers and $b$ is as small as possible.

Show full solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle \sqrt{75}\;= \sqrt{\textcolor{#010104}{25} \times 3}

Look for the largest perfect square that divides

75

. It is

25

, because

75 = 25 \times 3

\displaystyle \phantom{\sqrt{75}}\;= \textcolor{#010104}{\sqrt{25}} \times \sqrt{3}

Split the root:

\sqrt{xy} = \sqrt{x} \times \sqrt{y}

\displaystyle \phantom{\sqrt{75}}\;= \textcolor{#010106}{5\sqrt{3}}

The root of the perfect square comes out:

\sqrt{25} = 5

Answer

5\sqrt{3}

Example 2.2

A coefficient in front

Simplify $3\sqrt{32}$ .

Show solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle 3\sqrt{32}\;= 3\sqrt{\textcolor{#010104}{16} \times 2}

Look for the largest perfect square that divides

32

. It is

16

, because

32 = 16 \times 2

\displaystyle \phantom{3\sqrt{32}}\;= 3\textcolor{#010104}{\sqrt{16}} \times \sqrt{2}

Split the root:

\sqrt{xy} = \sqrt{x} \times \sqrt{y}

\displaystyle \phantom{3\sqrt{32}}\;= 3 \times 4\sqrt{2}

The root of the perfect square comes out:

\sqrt{16} = 4

\displaystyle \phantom{3\sqrt{32}}\;= \textcolor{#010106}{12\sqrt{2}}

Multiply the numbers outside the root.

Answer

12\sqrt{2}

Your Turn 2.2

Simplify $5\sqrt{28}$ .

Show full solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle 5\sqrt{28}\;= 5\sqrt{\textcolor{#010104}{4} \times 7}

Look for the largest perfect square that divides

28

. It is

4

, because

28 = 4 \times 7

\displaystyle \phantom{5\sqrt{28}}\;= 5\textcolor{#010104}{\sqrt{4}} \times \sqrt{7}

Split the root:

\sqrt{xy} = \sqrt{x} \times \sqrt{y}

\displaystyle \phantom{5\sqrt{28}}\;= 5 \times 2\sqrt{7}

The root of the perfect square comes out:

\sqrt{4} = 2

\displaystyle \phantom{5\sqrt{28}}\;= \textcolor{#010106}{10\sqrt{7}}

Multiply the numbers outside the root.

Answer

10\sqrt{7}

Example 2.3

Add and subtract like surds

Simplify $\sqrt{12} + \sqrt{27} - \sqrt{3}$ .

Show solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle \sqrt{12} + \sqrt{27} - \sqrt{3}\;= \sqrt{\textcolor{#010104}{4} \times 3} + \sqrt{\textcolor{#010104}{9} \times 3} - \sqrt{3}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{\sqrt{12} + \sqrt{27} - \sqrt{3}}\;= 2\sqrt{3} + 3\sqrt{3} - \sqrt{3}

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{\sqrt{12} + \sqrt{27} - \sqrt{3}}\;= \textcolor{#010105}{(2 + 3 - 1)\sqrt{3}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{\sqrt{12} + \sqrt{27} - \sqrt{3}}\;= \textcolor{#010106}{4\sqrt{3}}

Add the coefficients of the like surds.

Answer

4\sqrt{3}

Your Turn 2.3

Simplify $2\sqrt{8} + \sqrt{50} - 3\sqrt{2}$ .

Show full solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle 2\sqrt{8} + \sqrt{50} - 3\sqrt{2}\;= 2\sqrt{\textcolor{#010104}{4} \times 2} + \sqrt{\textcolor{#010104}{25} \times 2} - 3\sqrt{2}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{2\sqrt{8} + \sqrt{50} - 3\sqrt{2}}\;= 4\sqrt{2} + 5\sqrt{2} - 3\sqrt{2}

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{2\sqrt{8} + \sqrt{50} - 3\sqrt{2}}\;= \textcolor{#010105}{(4 + 5 - 3)\sqrt{2}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{2\sqrt{8} + \sqrt{50} - 3\sqrt{2}}\;= \textcolor{#010106}{6\sqrt{2}}

Add the coefficients of the like surds.

Answer

6\sqrt{2}

Example 2.4

Two different surds and a whole number

Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8} + \sqrt{25}$ .

Show solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle \sqrt{50} + \sqrt{18} - \sqrt{8} + \sqrt{25}\;= \sqrt{\textcolor{#010104}{25} \times 2} + \sqrt{\textcolor{#010104}{9} \times 2} - \sqrt{\textcolor{#010104}{4} \times 2} + \sqrt{\textcolor{#010104}{25}}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{\sqrt{50} + \sqrt{18} - \sqrt{8} + \sqrt{25}}\;= 5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} + 5

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{\sqrt{50} + \sqrt{18} - \sqrt{8} + \sqrt{25}}\;= \textcolor{#010105}{5 + (5 + 3 - 2)\sqrt{2}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{\sqrt{50} + \sqrt{18} - \sqrt{8} + \sqrt{25}}\;= \textcolor{#010106}{5 + 6\sqrt{2}}

Add the coefficients of the like surds.

Answer

5 + 6\sqrt{2}

Your Turn 2.4

Simplify $\sqrt{48} + 2\sqrt{12} + \sqrt{18} - \sqrt{49}$ .

Show full solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle \sqrt{48} + 2\sqrt{12} + \sqrt{18} - \sqrt{49}\;= \sqrt{\textcolor{#010104}{16} \times 3} + 2\sqrt{\textcolor{#010104}{4} \times 3} + \sqrt{\textcolor{#010104}{9} \times 2} - \sqrt{\textcolor{#010104}{49}}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{\sqrt{48} + 2\sqrt{12} + \sqrt{18} - \sqrt{49}}\;= 4\sqrt{3} + 4\sqrt{3} + 3\sqrt{2} - 7

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{\sqrt{48} + 2\sqrt{12} + \sqrt{18} - \sqrt{49}}\;= \textcolor{#010105}{-7 + 3\sqrt{2} + (4 + 4)\sqrt{3}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{\sqrt{48} + 2\sqrt{12} + \sqrt{18} - \sqrt{49}}\;= \textcolor{#010106}{-7 + 3\sqrt{2} + 8\sqrt{3}}

Add the coefficients of the like surds.

Answer

-7 + 3\sqrt{2} + 8\sqrt{3}

Example 2.5

Multiply surds

Simplify $3\sqrt{6}\times 2\sqrt{2}$ .

Show solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle 3\sqrt{6}\times 2\sqrt{2}\;= (3 \times 2)\sqrt{6 \times 2}

Multiply the numbers in front together and the numbers under the roots together.

\displaystyle \phantom{3\sqrt{6}\times 2\sqrt{2}}\;= 6\sqrt{12}

Work out each product.

\displaystyle \phantom{3\sqrt{6}\times 2\sqrt{2}}\;= 6\sqrt{\textcolor{#010104}{4} \times 3}

12

has a perfect-square factor

4

, so simplify.

\displaystyle \phantom{3\sqrt{6}\times 2\sqrt{2}}\;= \textcolor{#010106}{12\sqrt{3}}

Take the root of the perfect square out.

Answer

12\sqrt{3}

Your Turn 2.5

Simplify $-2\sqrt{3}\times 5\sqrt{6}$ .

Show full solutionHide solution

Colour keyperfect-square factor taken outfinal form or accepted answer

\displaystyle -2\sqrt{3}\times 5\sqrt{6}\;= (-2 \times 5)\sqrt{3 \times 6}

Multiply the numbers in front together and the numbers under the roots together.

\displaystyle \phantom{-2\sqrt{3}\times 5\sqrt{6}}\;= -10\sqrt{18}

Work out each product.

\displaystyle \phantom{-2\sqrt{3}\times 5\sqrt{6}}\;= -10\sqrt{\textcolor{#010104}{9} \times 2}

18

has a perfect-square factor

9

, so simplify.

\displaystyle \phantom{-2\sqrt{3}\times 5\sqrt{6}}\;= \textcolor{#010106}{-30\sqrt{2}}

Take the root of the perfect square out.

Answer

-30\sqrt{2}

Example 2.6

Divide surds

Simplify $\dfrac{12\sqrt{18}}{4\sqrt{3}}$ .

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle \dfrac{12\sqrt{18}}{4\sqrt{3}}\;= \dfrac{12}{4} \times \dfrac{\sqrt{18}}{\sqrt{3}}

Divide the numbers in front and the roots separately.

\displaystyle \phantom{\dfrac{12\sqrt{18}}{4\sqrt{3}}}\;= \dfrac{12}{4}\sqrt{6}

\sqrt{18} \div \sqrt{3} = \sqrt{18 \div 3} = \sqrt{6}

\displaystyle \phantom{\dfrac{12\sqrt{18}}{4\sqrt{3}}}\;= \textcolor{#010106}{3\sqrt{6}}

Cancel the fraction and simplify the root if you can.

Answer

3\sqrt{6}

Example 2.7

Which is larger?

Which is larger, $3\sqrt{5}$ or $4\sqrt{3}$ ? Write each as a single square root to decide.

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle 3\sqrt{5}\;= \sqrt{3^2 \times 5} = \sqrt{9 \times 5} = \textcolor{#010106}{\sqrt{45}}

Write

3

\sqrt{9}

and multiply under one root.

\displaystyle 4\sqrt{3}\;= \sqrt{4^2 \times 3} = \sqrt{16 \times 3} = \textcolor{#010106}{\sqrt{48}}

Do the same for the other number.

Both are now square roots, so compare the numbers under the roots:

45

and

48

. The larger root is

4\sqrt{3}

Answer

3\sqrt{5} = \sqrt{45}

and

4\sqrt{3} = \sqrt{48}

, so

4\sqrt{3}

is larger

Your Turn 2.7

Which is larger, $5\sqrt{2}$ or $3\sqrt{6}$ ? Write each as a single square root to decide.

Show full solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle 5\sqrt{2}\;= \sqrt{5^2 \times 2} = \sqrt{25 \times 2} = \textcolor{#010106}{\sqrt{50}}

Write

5

\sqrt{25}

and multiply under one root.

\displaystyle 3\sqrt{6}\;= \sqrt{3^2 \times 6} = \sqrt{9 \times 6} = \textcolor{#010106}{\sqrt{54}}

Do the same for the other number.

Both are now square roots, so compare the numbers under the roots:

50

and

54

. The larger root is

3\sqrt{6}

Answer

5\sqrt{2} = \sqrt{50}

and

3\sqrt{6} = \sqrt{54}

, so

3\sqrt{6}

is larger

Example 2.8

Exact answer, then a calculator check

Simplify $\sqrt{18} + \sqrt{32}$ exactly, then check your answer with a calculator.

Show solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle \sqrt{18} + \sqrt{32}\;= \sqrt{\textcolor{#010104}{9} \times 2} + \sqrt{\textcolor{#010104}{16} \times 2}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{\sqrt{18} + \sqrt{32}}\;= 3\sqrt{2} + 4\sqrt{2}

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{\sqrt{18} + \sqrt{32}}\;= \textcolor{#010105}{(3 + 4)\sqrt{2}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{\sqrt{18} + \sqrt{32}}\;= \textcolor{#010106}{7\sqrt{2}}

Add the coefficients of the like surds.

Calculator check: the original expression gives

9.8995

and the answer

7\sqrt{2}

gives

9.8995

, so they agree.

Compare the decimals; they must match.

Answer

7\sqrt{2}

Your Turn 2.8

Simplify $\sqrt{24} + 3\sqrt{54}$ exactly, then check your answer with a calculator.

Show full solutionHide solution

Colour keyperfect-square factor taken outlike surds collected, or parts matchedfinal form or accepted answer

\displaystyle \sqrt{24} + 3\sqrt{54}\;= \sqrt{\textcolor{#010104}{4} \times 6} + 3\sqrt{\textcolor{#010104}{9} \times 6}

Split each number under a root into a perfect square times what is left.

\displaystyle \phantom{\sqrt{24} + 3\sqrt{54}}\;= 2\sqrt{6} + 9\sqrt{6}

Take the square root of each perfect square, then multiply by the number in front.

\displaystyle \phantom{\sqrt{24} + 3\sqrt{54}}\;= \textcolor{#010105}{(2 + 9)\sqrt{6}}

Only like surds (the same number under the root) can be added or subtracted.

\displaystyle \phantom{\sqrt{24} + 3\sqrt{54}}\;= \textcolor{#010106}{11\sqrt{6}}

Add the coefficients of the like surds.

Calculator check: the original expression gives

26.9444

and the answer

11\sqrt{6}

gives

26.9444

, so they agree.

Compare the decimals; they must match.

Answer

11\sqrt{6}

Supports K341-A3-01

Expanding brackets that contain surds

Expanding with surds is ordinary expanding, followed by tidying the roots. Multiply every term in the first bracket by every term in the second, then use $\sqrt{a} \times \sqrt{a} = a$ and $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to simplify, and finally collect the rational parts and the like surds.

The Core Idea

Multiply every term by every term, then tidy the roots.

The three special products save time: $(x + y)^2$ , $(x - y)^2$ and $(x + y)(x - y)$ . The last one is the most useful, because the surds vanish.

Case table 3: the three special products
Name	Pattern	With surds
Square of a sum	$(x + y)^2 = x^2 + 2xy + y^2$	$(2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$
Square of a difference	$(x - y)^2 = x^2 - 2xy + y^2$	$(\sqrt{7} - \sqrt{3})^2 = 7 - 2\sqrt{21} + 3 = 10 - 2\sqrt{21}$
Difference of two squares	$(x + y)(x - y) = x^2 - y^2$	$(3 + \sqrt{5})(3 - \sqrt{5}) = 9 - 5 = 4$

Watch out.

(2 + \sqrt{3})^2

is not

4 + 3

. The middle term

2 \times 2 \times \sqrt{3}

is part of the answer, so squaring a bracket always has three terms.

SGSchoolKaki Tip

Spot a bracket pair like $(a + \sqrt{b})(a - \sqrt{b})$ at once: the answer is the whole number $a^2 - b$ , with no surd at all.

Example 3.1

A number times a bracket

Expand and simplify $3(2 + \sqrt{5})$ .

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle 3(2 + \sqrt{5})\;= 3 \times 2 + 3 \times \sqrt{5}

Multiply the number outside by every term in the bracket.

\displaystyle \phantom{3(2 + \sqrt{5})}\;= \textcolor{#010106}{6 + 3\sqrt{5}}

Work out each product.

Answer

6 + 3\sqrt{5}

Example 3.2

Two brackets

Expand and simplify $(2 + \sqrt{3})(1 - \sqrt{3})$ .

Show solutionHide solution

Colour keylike surds collected, or parts matchedfinal form or accepted answer

\displaystyle (2 + \sqrt{3})(1 - \sqrt{3})\;= 2 - 2 \times \sqrt{3} + \sqrt{3} - \sqrt{3} \times \sqrt{3}

Multiply every term in the first bracket by every term in the second.

\displaystyle \phantom{(2 + \sqrt{3})(1 - \sqrt{3})}\;= 2 - 2\sqrt{3} + \sqrt{3} - 3

Simplify each product: a root times itself is the number, and two roots multiply under one root.

\displaystyle \phantom{(2 + \sqrt{3})(1 - \sqrt{3})}\;= \textcolor{#010105}{(2 - 3) + (-2 + 1)\sqrt{3}}

Collect the whole numbers together and the like surds together.

\displaystyle \phantom{(2 + \sqrt{3})(1 - \sqrt{3})}\;= \textcolor{#010106}{-1 - \sqrt{3}}

Simplify.

Answer

-1 - \sqrt{3}

Your Turn 3.2

Expand and simplify $(3 + 2\sqrt{5})(2 - \sqrt{5})$ .

Show full solutionHide solution

Colour keylike surds collected, or parts matchedfinal form or accepted answer

\displaystyle (3 + 2\sqrt{5})(2 - \sqrt{5})\;= 3 \times 2 - 3 \times \sqrt{5} + 2\sqrt{5} \times 2 - 2\sqrt{5} \times \sqrt{5}

Multiply every term in the first bracket by every term in the second.

\displaystyle \phantom{(3 + 2\sqrt{5})(2 - \sqrt{5})}\;= 6 - 3\sqrt{5} + 4\sqrt{5} - 10

Simplify each product: a root times itself is the number, and two roots multiply under one root.

\displaystyle \phantom{(3 + 2\sqrt{5})(2 - \sqrt{5})}\;= \textcolor{#010105}{(6 - 10) + (-3 + 4)\sqrt{5}}

Collect the whole numbers together and the like surds together.

\displaystyle \phantom{(3 + 2\sqrt{5})(2 - \sqrt{5})}\;= \textcolor{#010106}{-4 + \sqrt{5}}

Simplify.

Answer

-4 + \sqrt{5}

Example 3.3

Square of a sum

Expand and simplify $(2 + 3\sqrt{5})^2$ .

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle (2 + 3\sqrt{5})^2\;= 2^2 + 2(2)(3\sqrt{5}) + (3\sqrt{5})^2

Use

(x \pm y)^2 = x^2 \pm 2xy + y^2

\displaystyle \phantom{(2 + 3\sqrt{5})^2}\;= 4 + 12\sqrt{5} + 45

Square each term:

(\sqrt{x})^2 = x

and

(k\sqrt{x})^2 = k^2x

\displaystyle \phantom{(2 + 3\sqrt{5})^2}\;= \textcolor{#010106}{49 + 12\sqrt{5}}

Work out the total.

Answer

49 + 12\sqrt{5}

Example 3.4

Square of two surds

Expand and simplify $(\sqrt{7} - \sqrt{3})^2$ .

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle (\sqrt{7} - \sqrt{3})^2\;= (\sqrt{7})^2 - 2(\sqrt{7})(\sqrt{3}) + (\sqrt{3})^2

Use

(x \pm y)^2 = x^2 \pm 2xy + y^2

\displaystyle \phantom{(\sqrt{7} - \sqrt{3})^2}\;= 7 - 2\sqrt{21} + 3

Square each term:

(\sqrt{x})^2 = x

and

(k\sqrt{x})^2 = k^2x

\displaystyle \phantom{(\sqrt{7} - \sqrt{3})^2}\;= \textcolor{#010106}{10 - 2\sqrt{21}}

Work out the total.

Answer

10 - 2\sqrt{21}

Example 3.5

The surds vanish

Expand and simplify $(4 + \sqrt{7})(4 - \sqrt{7})$ . What kind of number do you get?

Show solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle (4 + \sqrt{7})(4 - \sqrt{7})\;= 4^2 - (\sqrt{7})^2

Use

(x + y)(x - y) = x^2 - y^2

: the middle terms cancel.

\displaystyle \phantom{(4 + \sqrt{7})(4 - \sqrt{7})}\;= 16 - 7

Square each term:

(\sqrt{x})^2 = x

and

(k\sqrt{x})^2 = k^2x

\displaystyle \phantom{(4 + \sqrt{7})(4 - \sqrt{7})}\;= \textcolor{#010106}{9}

Work out the total.

Answer

9

, a rational number (no surd is left)

Your Turn 3.5

Expand and simplify $(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})$ . What kind of number do you get?

Show full solutionHide solution

Colour keyfinal form or accepted answer

\displaystyle (3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})\;= (3\sqrt{2})^2 - (2\sqrt{3})^2

Use

(x + y)(x - y) = x^2 - y^2

: the middle terms cancel.

\displaystyle \phantom{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})}\;= 18 - 12

Square each term:

(\sqrt{x})^2 = x

and

(k\sqrt{x})^2 = k^2x

\displaystyle \phantom{(3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3})}\;= \textcolor{#010106}{6}

Work out the total.

Answer

6

, a rational number (no surd is left)

Example 3.6

A longer second bracket

Expand and simplify $(1 + \sqrt{2})(1 + 2\sqrt{2} - \sqrt{3})$ .

Show solutionHide solution

Colour keylike surds collected, or parts matchedfinal form or accepted answer

\displaystyle (1 + \sqrt{2})(1 + 2\sqrt{2} - \sqrt{3})\;= 1 + 2\sqrt{2} - \sqrt{3} + \sqrt{2} + \sqrt{2} \times 2\sqrt{2} - \sqrt{2} \times \sqrt{3}

Multiply every term in the first bracket by every term in the second.

\displaystyle \phantom{(1 + \sqrt{2})(1 + 2\sqrt{2} - \sqrt{3})}\;= 1 + 2\sqrt{2} - \sqrt{3} + \sqrt{2} + 4 - \sqrt{6}

Simplify each product: a root times itself is the number, and two roots multiply under one root.

\displaystyle \phantom{(1 + \sqrt{2})(1 + 2\sqrt{2} - \sqrt{3})}\;= \textcolor{#010105}{(1 + 4) + (2 + 1)\sqrt{2} - \sqrt{3} - \sqrt{6}}

Collect the whole numbers together and the like surds together.

\displaystyle \phantom{(1 + \sqrt{2})(1 + 2\sqrt{2} - \sqrt{3})}\;= \textcolor{#010106}{5 + 3\sqrt{2} - \sqrt{3} - \sqrt{6}}

Simplify.

Answer

5 + 3\sqrt{2} - \sqrt{3} - \sqrt{6}

Your Turn 3.6

Expand and simplify $(2 - \sqrt{3})(1 + \sqrt{3} + 3\sqrt{2})$ .

Show full solutionHide solution

Colour keylike surds collected, or parts matchedfinal form or accepted answer

\displaystyle (2 - \sqrt{3})(1 + \sqrt{3} + 3\sqrt{2})\;= 2 + 2 \times \sqrt{3} + 2 \times 3\sqrt{2} - \sqrt{3} - \sqrt{3} \times \sqrt{3} - \sqrt{3} \times 3\sqrt{2}

Multiply every term in the first bracket by every term in the second.

\displaystyle \phantom{(2 - \sqrt{3})(1 + \sqrt{3} + 3\sqrt{2})}\;= 2 + 2\sqrt{3} + 6\sqrt{2} - \sqrt{3} - 3 - 3\sqrt{6}

Simplify each product: a root times itself is the number, and two roots multiply under one root.

\displaystyle \phantom{(2 - \sqrt{3})(1 + \sqrt{3} + 3\sqrt{2})}\;= \textcolor{#010105}{(2 - 3) + 6\sqrt{2} + (2 - 1)\sqrt{3} - 3\sqrt{6}}

Collect the whole numbers together and the like surds together.

\displaystyle \phantom{(2 - \sqrt{3})(1 + \sqrt{3} + 3\sqrt{2})}\;= \textcolor{#010106}{-1 + 6\sqrt{2} + \sqrt{3} - 3\sqrt{6}}

Simplify.

Answer

-1 + 6\sqrt{2} + \sqrt{3} - 3\sqrt{6}

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Notes version 1.0 · Last updated 3 Nov 2026 · Chapter 3 of 17