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Squares and square roots Simplifying surds Multiplying and dividing surdsThis topic is relevant for:

Here we will learn about **adding and subtracting surds** including when surd expressions can be added or subtracted, and how to carry out these calculations.

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

Adding and subtracting surds is where we can add or subtract surds when the numbers underneath the root symbols (the radicands) are the same; these are called ‘like surds’.

This is similar to collecting like terms in algebra:

E.g.

So when we do a similar thing with surds:

E.g.

\[\sqrt{3}+\sqrt{3}+2\sqrt{3}=4\sqrt{3}\]

And just as

2\sqrt{3}+3\sqrt{7} cannot be simplified because the numbers underneath the square root symbols (radicands) are different. \sqrt{3} and \sqrt{7} are not ‘like surds’.

If surds can be simplified so that they are ‘like surds’, then they can be added or subtracted.

You may be asked to apply these skills at GCSE maths to give answers to problems in geometry, such as Pythagoras or trigonometry, as exact values, rather than as decimals. Before calculators were invented, surds were the standard form for stating answers which were irrational numbers.

The formula for solving quadratics also uses a square root sign, so you may need to apply your knowledge of surds here as well.

In order to add and subtract surds:

**Check whether the terms are ‘like surds’.****If they aren’t like surds, simplify each surd as far as possible.****Combine the like surd terms by adding or subtracting.**

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\[2\sqrt{3}+5\sqrt{3}\]

**Check whether the terms are ‘like surds’.**

The number under the root sign is already

**2If they aren’t like surds, simplify each surd as far as possible.**

We don’t need to change the surd terms in this question.

**3Combine the like surd terms by adding or subtracting.**

\[2+5=7\]

so,

\[2\sqrt{3}+5\sqrt{3}=7\sqrt{3}\]

Simplify

\[7\sqrt{5}-9\sqrt{5}+6\sqrt{5}\]

**Check whether the terms are ‘like surds’.**

The number under the root sign is already

**If they aren’t like surds, simplify each surd as far as possible.**

We don’t need to change the surd terms in this question.

**Combine the like surd terms by adding or subtracting.**

\[7-9+6=4\]

so,

\[7\sqrt{5}-9\sqrt{5}+6\sqrt{5}=4\sqrt{5}\]

Simplify

\[10 \sqrt{2}+4 \sqrt{11}\]

**Check whether the terms are ‘like surds’.**

The numbers under the root signs are

**If they aren’t like surds, simplify each surd as far as possible.**

These surds cannot be simplified further. There are no square factors of either

**Combine the like surd terms by adding or subtracting.**

In this case, the surds are not like, so they cannot be combined.

We state the answer as:

\[10 \sqrt{2}+4 \sqrt{11}\]

Simplify

\[4+6\sqrt{2}-3\sqrt{2}-\sqrt{25}\]

**Check whether the terms are ‘like surds’.**

All the sure terms are like surds – they are all root

**If they aren’t like surds, simplify each surd as far as possible.**

We don’t need to simplify the surds.

However, the square root of

\[4+6\sqrt{2}-3\sqrt{2}-5\]

**Combine the like surd terms by adding or subtracting.**

Combining the surds:

\[6\sqrt{2}-3\sqrt{2}=3\sqrt{2}\]

We also combine the other like terms – in this case, the integers:

\[ 4-5=-1\]

So the final answer is:

\[3\sqrt{2}-1\]

Note that it’s always best to start an answer with a positive term, but:

\[-1+3\sqrt{2}\]

is also correct.

Simplify

\[\sqrt{7}+\sqrt{28}\]

**Check whether the terms are ‘like surds’.**

The numbers under the root signs are

**If they aren’t like surds, simplify each surd as far as possible.**

\[\sqrt{7}\]

is already fully simplified; there are no square factors of

\[\sqrt{28}\]

can be simplified, because

\[\begin{aligned}
\sqrt{28} &=\sqrt{4 \times 7} \\
&=\sqrt{4} \times \sqrt{7} \\
&=2 \times \sqrt{7} \\
&=2 \sqrt{7}
\end{aligned}\]

**Combine the like surd terms by adding or subtracting.**

\[\begin{aligned}
\sqrt{7}+\sqrt{28} &=\sqrt{7}+2 \sqrt{7} \\
&=3 \sqrt{7}
\end{aligned}\]

Simplify

\[\sqrt{8}+\sqrt{72}\]

**Check whether the terms are ‘like surds’.**

The numbers under the root signs are

**If they aren’t like surds, simplify each surd as far as possible.**

\[\sqrt{8}\]

can be simplified, because

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

\[\sqrt{72}\]

can be simplified, because

\[\begin{aligned}
\sqrt{72} &=\sqrt{36 \times 2} \\
&=\sqrt{36} \times \sqrt{2} \\
&=6 \times \sqrt{2} \\
&=6 \sqrt{2}
\end{aligned}\]

Note that you could have simplified in stages, using the square factors

\[\begin{aligned}
\sqrt{72} &=\sqrt{4} \times \sqrt{18} \\
&=2 \sqrt{18}
\end{aligned}\]

Then the surds would not be ‘like’ and you would not be able to combine by addition or subtraction.

**Combine the like surd terms by adding or subtracting.**

\[\begin{aligned}
\sqrt{8}+\sqrt{72} &=2\sqrt{2}+6 \sqrt{2} \\
&=8 \sqrt{2}
\end{aligned}\]

Simplify

\[\sqrt{75}+\sqrt{50}\]

**Check whether the terms are ‘like surds’.**

The numbers under the root signs are

**If they aren’t like surds, simplify each surd as far as possible.**

\[\sqrt{75}\]

can be simplified, because

\[\begin{aligned}
\sqrt{75} &=\sqrt{25 \times 3} \\
&=\sqrt{25} \times \sqrt{3} \\
&=5 \times \sqrt{3} \\
&=5 \sqrt{3}
\end{aligned}\]

\[\sqrt{50}\]

can be simplified, because

\[\begin{aligned}
\sqrt{50} &=\sqrt{25 \times 2} \\
&=\sqrt{25} \times \sqrt{2} \\
&=5 \times \sqrt{2} \\
&=5 \sqrt{2}
\end{aligned}\]

**Combine the like surd terms by adding or subtracting.**

Even when simplified fully, the surds are not like, so they cannot be combined.

We state the answer as:

\[5\sqrt{3}+5\sqrt{2}\]

Simplify

\[\sqrt{90}+\sqrt{64}-\sqrt{40}\]

**Check whether the terms are ‘like surds’.**

The numbers under the root signs are

**If they aren’t like surds, simplify each surd as far as possible.**

\[\sqrt{90}\]

can be simplified, because

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

\[\sqrt{64}=8\]

\[\begin{aligned}
\sqrt{40} &=\sqrt{4 \times 10} \\
&=\sqrt{4} \times \sqrt{10} \\
&=2 \times \sqrt{10} \\
&=2 \sqrt{10}
\end{aligned}\]

**Combine the like surd terms by adding or subtracting.**

\[\begin{aligned}
\sqrt{90}+\sqrt{64}-\sqrt{40} &=3 \sqrt{10}+8-2 \sqrt{10} \\
&=8+\sqrt{10}
\end{aligned}\]

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

As in algebra, we understand that

**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:

^{2}

**Not simplifying each surd fully**

As in example

**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.

Adding and subtracting surds is part of our series of lessons to support revision on surds. You may find it helpful to start with the main surds lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

1. Simplify:

3 \sqrt{11}+2 \sqrt{11}-\sqrt{11}

6 \sqrt{11}

4 \sqrt{11}

7 \sqrt{11}

5 \sqrt{11}

Already in like surd form; 3+2-1=4 .

2. Simplify:

3\sqrt{5}-2\sqrt{3}+4\sqrt{3}

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

5 \sqrt{15}

-24 \sqrt{15}

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

We can only collect the ‘like’ root 3 s; 4-2=2 .

3. Simplify:

\sqrt{11}+\sqrt{44}

5\sqrt{11}

\sqrt{55}

3\sqrt{11}

11 \sqrt{5}

4 is a square factor of 44 , so use this to simplify root 44 . There’s then a like surd of root 11 .

4. Simplify:

\sqrt{54}-\sqrt{24}

\sqrt{30}

6\sqrt{30}

6\sqrt{6}

\sqrt{6}

9 is a square factor of 54 and 4 is a square factor of 24 . When you simplify both roots, there’s a like surd of root 6 .

5. Simplify:

3\sqrt{20}-\sqrt{50}

3\sqrt{1000}

6-\sqrt{10} \sqrt{5}

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

6\sqrt{10} \sqrt{5}

When both surds are fully simplified, they are not like surds. We just write the answer as the subtraction with simplified surds.

6. Simplify:

3 \sqrt{16}+\sqrt{50}-\sqrt{8}

12+5 \sqrt{10}-2 \sqrt{2}

12+3 \sqrt{2}

\sqrt{2}(3+3 \sqrt{8})

12-3 \sqrt{2}

Root 16 is not a surd, so we work out 3\times4=12 . Then, looking at the surds, when both are simplified, there is a like surd of root 2 . We combine these two surds.

1. Express \sqrt{6}+\sqrt{54} in the form a \sqrt{6} where a is an integer.

**(2 marks)**

Show answer

\sqrt{54}=\sqrt{9} \times \sqrt{6}

**(1)**

\sqrt{6} + \sqrt{54} =4\sqrt{6} (a=4)

**(1)**

2. Simplify fully \sqrt{32}+\sqrt{2}

**(2 marks)**

Show answer

\sqrt{32}=\sqrt{16} \times \sqrt{2}

**(1)**

5 \sqrt{2}

**(1)**

3. Write \sqrt{40}+\sqrt{160} in the form a \sqrt{10}

**(3 marks)**

Show answer

\sqrt{40}=\sqrt{4} \times \sqrt{10}

**(1)**

\sqrt{160}=\sqrt{16} \times \sqrt{10}

**(1)**

\sqrt{40} + \sqrt{160} =6\sqrt{10} (a=6)

**(1)**

You have now learned how to:

- Add and subtract surds

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