# Adding surds – What students get wrong about expressions with square roots

Colin Foster looks at the different ways in which surds can be combined – some of which can be difficult for students to make sense of…

- by Colin Foster

**In this lesson, students contrast multiplication and addition of surds to understand how they are different but related.**

## The difficulty – adding and subtracting surds

Look at these statements. Are they true or false? Why?

√12 + √3 = √15

√12 – √3 = √9

√12 × √3 = √36

√12 ÷ √3 = √4

Students may need calculators to be sure. Three of the right-hand sides are square roots of perfect squares, so students may recognise these integers (√36 = 6, √9 = 3, √4 = 2).

The multiplication and the division are correct, but the addition and the subtraction aren’t.

## The solution

How can we be sure whether these are true or false?

We can show that √12 × √3 must equal √36, because squaring each of these expressions gives us the same value:

( √12 √3 )( √12 √3 ) =? √36 √36

( √12 √12 )( √3 √3 ) =? √36 √36

12 × 3 =? 36

After squaring, the left-hand side equals the right-hand side, meaning that √12 × √3 = √36.

## Adding surds with a calculator

If we try this with the ** addition**, we get a problem:

√12 + √3 =? √15

( √12 + √3 ) 2 =? √15 √15

( √12 ) 2 + 2√12√3 + ( √3 ) 2 =? √15√15

12 + 2√12√3 + 3 =? 15

So, we can see not only that these are ** not equal** but that the left-hand side in a case like this is

**going to be bigger (because of the extra 2√12√3 term, which**

*always***).**

*must be positive*o, √12 + √3 > √12 + 3, ** not** √12 + √3 = √12 + 3.

Square rooting is ** sub-additive**, which means that, unless either

*a*or

*b*is zero, √

*a*+

*b*< √

*a*+ √

*b*.

The radical symbol √ does ** not** behave like multiplication – e.g. something like 3(

*a*+

*b*) = 3

*a*+ 3

*b*).

Square rooting is ** not distributive over addition** like multiplication is.

However, we ** can** simplify √12 + √3, by using what we have seen about

**of surds.**

*multiplication*The number 12 has a ** square factor** (4), and so we can write √12 = √4√3, and because 4 is a square number, √12 is equal to 2√3.

So, √12 + √3 = 2√3 + √3 = 3√3.

(This last step is just ‘counting in √3s’ and is analogous to collecting like terms.)

Students should check this on their calculators. We can simplify additions and subtractions of surds in this way whenever there is a square number that is a factor of the number being square rooted.

*What would √12 − √3 be equal to?*

This time, √12 − √3 = 2√3 − √3 = √3. It may look strange to write √12 − √3 = √3 (students may think it should be √6 − √3 = √3), but it is correct.

## Checking for understanding

To assess students’ understanding, ask them to find as many surd expressions as they can that are equal to 3√5. For example, they could start with 4√5 − √5 and convert this to √80 − √5.

There are many possibilities, and generating lots of these is an excellent way to practise using these ideas. They could try to make the 3√5 as concealed as possible, and make some that look as though they ** might** be equal to 3√5

**!**

*but aren’t***Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk**