Adding and subtracting surds — like terms only — Mathematics

Before this

Simplifying surds
Collecting like terms · coming soon
Taking out a common factor · coming soon

This leads to

Two sums that look alike, and behave differently

Two square tiles sit side by side on a floor. One has area 8 cm². The other has area 18 cm². How wide are they together?

The first tile has side $8$ cm and the second has side $18$ cm. So the total width is $8 + 18$ cm. Many students now add the numbers inside and write $26$ . A calculator says no. $26 \approx 5.10$ , but $8 + 18 \approx 7.07$ .

Here is the surprise. $7.07$ is not a random decimal. It is $50$ , and $50 = 52$ . So this sum does collapse into a single surd. It is just not the one most people guess.

Now try $2 + 3$ . The calculator gives about 3.146. That is not $5$ , and it is not 5 times anything simple. This sum does not collapse at all.

So why does one sum of surds become a single surd, and another refuses? The answer is one idea you already use in algebra. This lecture shows that idea, proves it, and then shows why the second sum really is stuck.

The idea you already know: like terms

You know that $3 x + 5 x = 8 x$ . Why is that true? It is not a rule about the letter $x$ . It is the distributive law, used backwards. The common factor $x$ comes out of both terms:

3 x + 5 x = (3 + 5) x = 8 x

This works for any number $x$ . So it works when $x$ is $2$ . That means $32 + 52 = (3 + 5) 2 = 82$ . We are not using a new rule for surds. We are using the same law that collects like terms in algebra.

Look at it as lengths. Think of $2$ as a rod of one fixed length. Then $22$ is two of those rods end to end, and $32$ is three. Put them together and you have five rods.

Top: rods of the same length simply count up, so 2√2 + 3√2 = 5√2. Bottom: a √2 rod and a √3 rod together do not end at a whole number of either kind of rod.

This is why the number under the root matters so much. Rods of the same length can be counted. Rods of different lengths cannot be counted as one kind of rod.

Adding and subtracting surds

Surds with the same number under the root are called like surds. Like surds combine by adding or subtracting the numbers in front. The root does not change.

$a n + b n = (a + b) n$ and $a n - b n = (a - b) n$

Surds with different numbers under the root, in simplest form, are unlike surds. They stay as separate terms.

The number under the root has a name: the radicand. The number in front is the coefficient, the same word as in algebra. A surd with nothing in front, such as $5$ , has coefficient 1, just as $x$ means $1 x$ .

Surds in disguise: simplify first

Go back to the tiles. $8$ and $18$ have different radicands, so they look like unlike surds. But the rule above says "in simplest form". Simplify each one, using the largest square factor from the last lecture:

8 + 18 = 4 \times 2 + 9 \times 2 = 22 + 32 = 52

That is it. That is the whole idea. Both surds were secretly multiples of $2$ , and simplifying showed it. The tiles are $52 \approx 7.07$ cm wide together, which matches the calculator.

The last lecture promised that this would matter. It showed that $72$ , $50$ and $8$ are $62$ , $52$ and $22$ . As written, $72 + 50 - 8$ looks hopeless. In simplest form it is $62 + 52 - 22 = 92$ . You cannot see that a match exists until every surd is in simplest form. So simplifying always comes first.

Before reading on, commit to an answer below. It tests the mistake this lecture is built to prevent.

Predict, then reveal — what is √5 + √7?

Choose the answer you believe. Then read why each choice is right or wrong.

The value of √5 + √7 on a number line, compared with two popular wrong answers:

√5 + √7 ≈ 4.88. It is not √12 (≈ 3.46), which comes from adding the numbers inside. It is not √35 (≈ 5.92), which is √5 × √7. The radicands 5 and 7 have no square factors, so √5 + √7 is already in simplest form.

If you chose $12$ , you are in good company. It is the most common mistake in this topic. Square roots join under multiplication, $a \times b = ab$ . They do not join under addition.

Why √2 + √3 really is stuck

A careful student should ask one more question. Maybe $2 + 3$ equals some single surd that nobody has found yet, for example $k m$ with a strange $k$ and $m$ . The rods suggest not, but a picture is not a proof. Here is the proof. This is the hard part of the lecture. Spend time here.

Suppose $2 + 3 = k m$ , where $k$ is a fraction (a whole number counts as a fraction too) and $m$ is a whole number. Square both sides. On the left, use $(p + q)^{2} = p^{2} + 2 pq + q^{2}$ :

(2 + 3)^{2} = 2 + 26 + 3 = 5 + 26

On the right, $(k m)^{2} = k^{2} m$ . That is a fraction, because $k$ is a fraction and $m$ is a whole number. So $5 + 26$ would be a fraction. Then rearranging gives $6 = (k^{2} m - 5) \div 2$ , which is also a fraction.

But $6$ is not a fraction. In the first lecture of this topic you saw that the square root of a whole number that is not a perfect square is irrational, and 6 is not a perfect square. So the starting assumption must be false. $2 + 3$ cannot be written as a single surd. Leaving it as two terms is not laziness. It is the exact, simplest answer.

Stop and think

A student writes $23 + 32 = 55$ . Another writes $23 + 32 = 56$ . Which one is right?

Show the answer

Neither. The radicands are 3 and 2. Both surds are already in simplest form, and the radicands are different, so these are unlike surds. The answer stays as $23 + 32$ . Check: $23 + 32 \approx 3.46 + 4.24 = 7.71$ , while $55 \approx 11.18$ and $56 \approx 12.25$ . The numbers in front do not decide whether surds are like. Only the radicand does.

Worked examples

Worked Example 1 — Like surds, including a hidden 1

Simplify $65 - 25 + 5$ .

All three terms have radicand 5, so they are like surds. The last term has coefficient 1. Collect the coefficients:

65 - 25 + 15 = (6 - 2 + 1) 5 = 55

Check: $65 - 25 + 5 \approx 13.416 - 4.472 + 2.236 = 11.180$ , and $55 \approx 11.18$ .

Worked Example 2 — A mix of like and unlike surds

Simplify $32 + 43 + 8 - 12$ .

Step 1: simplify every surd. $8 = 22$ and $12 = 4 \times 3 = 23$ . The expression becomes $32 + 43 + 22 - 23$ .

Step 2: group the like surds. Keep each sign attached to its term.

(32 + 22) + (43 - 23) = 52 + 23

Step 3: $2$ and $3$ are unlike, so $52 + 23$ is the final answer. Check: both forms are about 10.54.

Worked Example 3 — An answer below zero

Simplify $20 - 45$ .

Simplify each surd: $20 = 4 \times 5 = 25$ and $45 = 9 \times 5 = 35$ . So

20 - 45 = 25 - 35 = (2 - 3) 5 = - 5

A square root itself is never negative. But a difference of two roots can be. Here the second root is bigger. Check: $4.47 - 6.71 = - 2.24$ , and $- 5 \approx - 2.24$ .

Worked Example 4 — An exam-style perimeter

A rectangle has sides $50$ cm and $18$ cm. Find its perimeter in the form $k 2$ .

The form $k 2$ is a clue from the examiner: every side must turn out to be a multiple of $2$ . Simplify: $50 = 52$ and $18 = 32$ .

Perimeter = 2 (52 + 32) = 2 \times 82 = 162 cm

So $k = 16$ . Check: $162 \approx 22.63$ cm, and $2 (50 + 18) \approx 2 (7.071 + 4.243) \approx 22.63$ cm.

Now practise the decision itself. For each sum below, simplify first. Then decide whether it becomes one surd or stays as two terms.

Sorter — one surd, or two terms?

Choose a card, then choose the box it belongs in. Each card explains itself when you place it.

Each sum, simplified, and where it belongs:

Sum	Simplest form	Result
$8 + 18$	$22 + 32 = 52$	one surd
$10 + 15$	already simplest	two terms
$48 + 75$	$43 + 53 = 93$	one surd
$23 + 32$	already simplest	two terms
$6 + 24$	$6 + 26 = 36$	one surd
$50 - 32$	$52 - 42 = 2$	one surd
$18 + 12$	$32 + 23$	two terms
$27 - 3$	$33 - 3 = 23$	one surd

Did you notice that the cards that looked alike were not always the ones that combined? $18 + 12$ and $8 + 18$ look almost the same, but only one of them collapses. You can only tell after simplifying.

Three classic mistakes

Trap 1 — Adding the numbers inside the roots

$5 + 7$ is not $12$ . The quickest test is to square a small case: $9 + 16 = 3 + 4 = 7$ , but $25 = 5$ . Roots join under multiplication only.

Trap 2 — Changing the root while collecting

$32 + 52 = 82$ , not $84$ . You add the coefficients, just as $3 x + 5 x = 8 x$ and not $8 x^{2}$ . The radicand stays exactly as it was.

Trap 3 — Deciding before simplifying

$8 + 18$ looks like two unlike surds, and many students stop there. In simplest form it is $22 + 32 = 52$ . Never decide "these cannot combine" until every surd is in simplest form.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

Simplify $73 - 23 + 43$ and $57 - 7$ .

Hint 1
In each expression, every term has the same radicand. So you only work with the numbers in front.

Hint 2
$7$ on its own has coefficient 1.

Full answer
$73 - 23 + 43 = (7 - 2 + 4) 3 = 93$ . $57 - 7 = (5 - 1) 7 = 47$ . Check: $93 \approx 15.59$ and $47 \approx 10.58$ , which match the original expressions on a calculator.
Simplify $98 + 32$ and $75 - 12$ .

Hint 1
Simplify each surd first. Look for the largest square factor: 49, 16, 25 and 4 all appear.

Hint 2
$98 = 49 \times 2$ and $75 = 25 \times 3$ .

Full answer
$98 + 32 = 72 + 42 = 112$ . $75 - 12 = 53 - 23 = 33$ . Check: $112 \approx 15.56$ and $9.90 + 5.66 = 15.56$ . Also $33 \approx 5.20$ and $8.66 - 3.46 = 5.20$ .
Simplify $245 - 20 + 80$ .

Hint 1
Simplify each root. In the first term, the 2 outside then multiplies the number that comes out.

Hint 2
$45 = 35$ , so $245 = 65$ . Now do $20$ and $80$ .

Full answer
$245 = 2 \times 35 = 65$ . $20 = 25$ . $80 = 16 \times 5 = 45$ . So the expression is $65 - 25 + 45 = 85$ . Check: $85 \approx 17.89$ , and $13.42 - 4.47 + 8.94 = 17.89$ .
A right-angled triangle has shorter sides $2$ cm and $8$ cm. Find its perimeter in simplest surd form. Explain why your answer has two terms.

Hint 1
First find the hypotenuse with Pythagoras' theorem. Square each shorter side, add, then take the square root.

Hint 2
The hypotenuse is $2 + 8 = 10$ . Now add all three sides, simplifying $8$ first.

Full answer
Hypotenuse: $c^{2} = (2)^{2} + (8)^{2} = 2 + 8 = 10$ , so $c = 10$ cm. Perimeter: $2 + 8 + 10 = 2 + 22 + 10 = 32 + 10$ cm. The answer has two terms because $10$ is already in simplest form (10 has no square factor), and its radicand is not 2. So $10$ and $2$ are unlike surds. Check: $32 + 10 \approx 4.24 + 3.16 = 7.40$ cm, and $1.41 + 2.83 + 3.16 = 7.40$ cm.

The bigger idea

Like surds behave like measurements in the same unit. You can add 3 metres and 5 metres to get 8 metres. You cannot add 3 metres and 5 seconds into one number. In the same way, $2$ and $3$ act like two different units, and the proof above shows that no exchange rate between them exists.

This is one idea appearing in many places: things combine only when they are the same kind of thing. It decides which terms collect in algebra, which units add in physics, and, later, which directions add in vectors. Simplifying first is how you find out what kind of thing each term really is.

Key concepts in this lesson

Like surds combine by adding or subtracting their coefficients, $a n + b n = (a + b) n$ , which is the distributive law used to collect like terms.
Two surds are like surds only when, in simplest form, they have the same radicand, so every surd must be simplified before you decide.
A surd with no number in front, such as $5$ , has coefficient 1.
Square roots do not join under addition, so $a + b$ is not $a + b$ .
Unlike surds such as $2 + 3$ cannot be written as one surd, because squaring would make $6$ a fraction.