# Surds

Here we will learn about surds, including simplifying surds, adding and subtracting surds, multiplying surds and dividing surds and rationalising surds.

There are also surds worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are surds?

Surds are numbers left as square roots that give irrational numbers.

An irrational number can’t be written as a fraction, and in decimal form is infinitely long with no recurring pattern – they would go on for ever.

Surds can be a square root, cube root, or other root and are used when detailed accuracy is required in a calculation.

For example the square root of 3 and the cube root of 2 are both surds.

For Example

\sqrt{5} \approx 2.23606 , which is an irrational number.

The square root of 5 is a surd.

## Surds and exact calculations

When we are not able to simplify a number to remove a square root then it is a surd (irrational number). We use surds to write numbers precisely.

The decimal form of irrational numbers do not terminate or recur, so cannot be written exactly in decimal form. We can leave these numbers written as a surd which represents the value in its exact form.

For example,

\sqrt2 = 1.4142135623 ...

If we were to use 1.41 (2dp) in our calculations they would not be accurate. We need to use it in exact form, \sqrt2, so that the full value of the number is used.

We may also come across questions that ask us to leave our answers in exact form.

For example,

We need to use Pythagoras’ theorem.

a^{2}+b^{2} =c^{2}

4^{2}+6^{2} =x^{2}

16+36 =x^{2}

52 =x^{2}

x =\sqrt{52}

x =2 \sqrt{13}

## Rational and irrational numbers

### Rational numbers

A number that can be written as an integer (whole number) or a simple fraction is called a rational number. Rational numbers can be terminating decimals  or recurring decimals.

E.g.

2, 100, -3, \frac{3}{4} and \frac{1}{9}.

Any number that can’t be written in this form is called an irrational number.

### Irrational numbers

Irrational numbers in decimal form are infinite, with no recurring or repeating pattern.

E.g.

\pi is an example of an irrational number

When a root (square root, cube root or higher) gives an irrational number, it is called a surd. At GCSE, we are only concerned with square roots.

E.g.

\sqrt{4}=2 , which is an integer,

The square root of 4 is not a surd.

\sqrt{5} \approx 2.23606 , which is an infinitely long decimal with no recurring or repeating pattern, i.e. an irrational number.

The square root of 5 is a surd.

## 1. Simplifying surds

### Example: simplifying a surd

Simplify:

\sqrt{60}

1. Find a square number that is a factor of the number under the root.

Square numbers are 1, 4, 9, 16, 25,

4 is a factor of 60 (because 4 \times 15=60 ).

2Rewrite the surd as a product of this square number and another number, then evaluate the root of the square number.

\begin{aligned} \sqrt{60} &=\sqrt{4 \times 15} \\\\ &=\sqrt{4} \times \sqrt{15} \\\\ &=2 \times \sqrt{15} \\\\ &=2 \sqrt{15} \end{aligned}

3Repeat if the number under the root still has square factors.

In this case, there are no square numbers that are factors of 2 , so the surd is fully simplified.

\sqrt{60}=2\sqrt{15}

Step by step guide: Simplifying surds

## 2. Adding and subtracting surds

Simplify:

2\sqrt{5}+\sqrt{45}

The numbers under the root signs are 5 and 45 ; these are not like surds.

2\sqrt{5} is already fully simplified; there are no square factors of 5 .

\sqrt{45} can be simplified, because 9 is a square factor of 45 .

\begin{aligned} \sqrt{45} &=\sqrt{9 \times 5} \\\\ &=\sqrt{9} \times \sqrt{5} \\\\ &=3 \times \sqrt{5} \\\\ &=3 \sqrt{5} \end{aligned}

\begin{aligned} 2 \sqrt{5}+\sqrt{45} &=2 \sqrt{5}+3 \sqrt{5} \\\\ &=5 \sqrt{5} \end{aligned}

Step by step guide: Adding and subtracting surds

## 3. Multiplying and dividing surds

### Example: simplifying surd fractions

\frac{2 \sqrt{10}-\sqrt{90}}{\sqrt{2}}

There are no square factors of 10 , so \sqrt{10} cannot be simplified further.

9 is a square factor of 90 , so \sqrt{90} can be simplified as follows:

\begin{aligned} \sqrt{90} &=\sqrt{9 \times 10} \\\\ &=\sqrt{9} \times \sqrt{10} \\\\ &=3 \sqrt{10} \end{aligned}

There are no square factors of 2 , so \sqrt{2} cannot be simplified further.

The expression becomes:

\frac{2 \sqrt{10}-3\sqrt{10}}{\sqrt{2}}

The numerator contains like surds, so it can be simplified further.

\frac{2\sqrt{10}-3\sqrt{10}}{\sqrt{2}}=\frac{-1\sqrt{10}}{\sqrt{2}} .

As when simplifying algebraic expressions, we would usually write -1\sqrt{10} as just -\sqrt{10}.

-\sqrt{10}\div\sqrt{2}=-\sqrt{5}

-\sqrt{5}

Step by step guide: Multiplying and dividing surds

### Example: expanding a bracket with surds

Expand and simplify:

\sqrt{3}(6+\sqrt{8})

There are no square factors of 3 , so \sqrt{3} cannot be simplified further.

4 is a square factor of 8 , so \sqrt{8} can be simplified as follows:

\begin{aligned} \sqrt{8}&=\sqrt{4 \times 2}\\\\ &=\sqrt{4} \times \sqrt{2}\\\\ &=2\sqrt{2} \end{aligned}

So the expression becomes \sqrt{3}(6+2 \sqrt{2}) .

You can do this in a table, as you would if you were expanding brackets containing algebraic terms.

\sqrt{3} \times 6=6 \sqrt{3}

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

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

This cannot be simplified further.

## 4. Rationalise the denominator

### Example: rationalising the denominator

\frac{12}{\sqrt{54}}

Root 54 will simplify:

\begin{aligned} \sqrt{54} &=\sqrt{9 \times 6} \\\\ &=\sqrt{9} \times \sqrt{6} \\\\ &=3 \sqrt{6} \end{aligned}

\frac{12}{\sqrt{54}}=\frac{12}{3 \sqrt{6}}

So here we multiply the top and the bottom of the fraction by root 6.

\frac{12 \times \sqrt{6}}{3 \sqrt{6} \times \sqrt{6}}

Numerator:

12\times\sqrt{6}=12\sqrt{6}

Denominator:

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

So the full expression becomes:

\frac{12 \sqrt{6}}{18}

The denominator is now rationalised, because 18 is a rational number.

12\div 18=\frac{12}{18}=\frac{2}{3}

So the final answer simplifies to

\frac{2 \sqrt{6}}{3}

Step by step guide: Rationalising the denominator

## 5. Rationalising surds

### Example: rationalising using the conjugate of the denominator

Rationalise the denominator:

\frac{6}{\sqrt{11}-3}

The denominator is

(\sqrt{11}-3)

so we need

(\sqrt{11}+3)

\frac{6 \times(\sqrt{11}+3)}{(\sqrt{11}-3)(\sqrt{11}+3)}

Numerator:

6 \times(\sqrt{11}+3)=6 \sqrt{11}+18

Denominator:

\begin{aligned} (\sqrt{11}-3)(\sqrt{11}+3) &=11+3 \sqrt{11}-3 \sqrt{11}-9 \\\\ &=11-9 \\\\ &=2 \end{aligned}

So the full expression becomes:

\frac{6 \sqrt{11}+18}{2}

The denominator is now rationalised, because 2 is a rational number.

2 is a factor of 6 and 18, so we can divide through by 2 , leaving the answer as:

3 \sqrt{11}+9

Step-by-step guide: Rationalise the denominator

### Common misconceptions

• Incorrectly rewriting the number under the square root sign (the radicand) as a product of any two factors

One of these factors must be a square number in order for you to be able to simplify the surd.

• Not simplifying fully

Always check that there are no square factors of the number under the root.

• Remember that a root with no integer coefficient is ‘ 1 lot’ of that surd

As in algebra, we understand that a actually means 1a .

• Don’t mix up addition and multiplication laws
\sqrt{3}+\sqrt{3}=2 \sqrt{3}

\sqrt{3} \times \sqrt{3}=\sqrt{9}=3

Refer back to knowledge of algebra to help:

a+a=2a and a \times a=a^2.

• Trying to combine unlike surds

It’s OK to leave an answer with more than one surd in it if it will not simplify further.

• Take care when expanding double brackets when using a conjugate surd expression

Remember to multiply everything in the first bracket by everything in the second, and check that you have four terms before simplifying.

### Practice surds questions

1. Simplify fully:

\sqrt{450}

3 \sqrt{50}

8 \sqrt{2}

15 \sqrt{2}

9 \sqrt{50}

In one stage:

\sqrt{450}=\sqrt{225 \times 2}=15 \sqrt{2}

In two stages:

9 \times 50=450, so rewrite \sqrt{450}=\sqrt{9 \times 50}=3 \sqrt{50}

25 \times 2=50, so rewrite 3 \sqrt{50}=3 \sqrt{25 \times 2}=3 \times 5 \times \sqrt{2}=15 \sqrt{2}

2. Simplify:

\sqrt{50}-2 \sqrt{2}

2 \sqrt{25}

3 \sqrt{2}

2 \sqrt{5}

2 \sqrt{48}

Rewrite \sqrt{50}=\sqrt{25 \times 2}=5 \sqrt{2} , then subtract like surds to get the answer.

3. Simplify fully:

\frac{\sqrt{60}}{2 \sqrt{3}}

\frac{\sqrt{5}}{2}

\frac{\sqrt{20}}{2}

2 \sqrt{20}

\sqrt{5}

Simplify \sqrt{60}=\sqrt{4 \times 15}=2 \sqrt{15}

2\sqrt{15}\div 2\sqrt{3}=\sqrt{5}

4. Expand and simplify:

4(3+\sqrt{28})

12+2\sqrt{7}

12+4\sqrt{28}

12+4\sqrt{7}

12+8\sqrt{7}

Simplify \sqrt{28}=\sqrt{4 \times 7}=2 \sqrt{7}

\begin{aligned} &4 \times 3=12 \\\\ &4 \times 2 \sqrt{7}=8 \sqrt{7} \end{aligned}

So the final answer is 12+8\sqrt{7}.

5. Expand and simplify:

(2+\sqrt{3})(2-\sqrt{3})

4-3 \sqrt{3}

1

-1

7

Multiply each term in the first bracket by each term in the second bracket to get 4+2 \sqrt{3}-2 \sqrt{3}-3. The surd terms cancel out (as with a difference of two squares), and the integer terms simplify to 1.

6. Rationalise:

\frac{4}{\sqrt{5}}

\frac{\sqrt{5}}{5}

4 \sqrt{5}

\frac{4 \sqrt{5}}{5}

Multiply the numerator and denominator by \sqrt{5}.

### Surds GCSE questions

1. \sqrt{775}=5\sqrt{k}

where k is an integer. Find the value of k.

(2 marks)

\sqrt{775}=\sqrt{25}\times\sqrt{k}

or sight of 775 \div 25

(1)

5\sqrt{31}

so k=31

(1)

2. (a) Simplify \sqrt{375}

(b) Hence, or otherwise, simplify:

\sqrt{375}+\sqrt{960}

(4 marks)

(a)

\sqrt{375}=\sqrt{25}\times\sqrt{15}

(1)

5 \sqrt{15}

(1)

(b)

\sqrt{960}=\sqrt{64} \times \sqrt{15}

(1)

13\sqrt{15}

(1)

3. (a) Expand and simplify

(5+\sqrt{7})(5-\sqrt{7})

(b) Hence, or otherwise, show that

\frac{36}{5+\sqrt{7}} can be written as 10-2 \sqrt{7}

(5 marks)

(a)

25+5 \sqrt{7}-5 \sqrt{7}-7

Any two correct terms

(1)

All four correct terms

(1)

Simplified to 18

(1)

(b)

\frac{36(5-\sqrt{7})}{(5+\sqrt{7})(5-\sqrt{7})}

(1)

\frac{180-36 \sqrt{7}}{18} leading to 10-2 \sqrt{7} as required

(1)

## Learning checklist

You have now learned how to:

• Simplify surds
• Multiply and divide surds
• Rationalise the denominator
• Rationalise other surd expressions

## The next lessons are

• Using exact values in trigonometry
• Geometric progressions with surds

## Still stuck?

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#### FREE GCSE maths practice papers (Edexcel, AQA & OCR)

8 sets of free exam practice papers written by maths teachers and examiners for Edexcel, AQA and OCR.

Each set of exam papers contains the three papers that your students will expect to find in their GCSE mathematics exam.