Simplifying surds — finding the largest square factor — Mathematics

Before this

What is a surd
Prime numbers · coming soon

This leads to

Two answers that are secretly the same

In the last lecture, a square garden had an area of 20 m², so its side was $20$ m. That is exact, and it is correct.

Now suppose a second student measures the same garden and writes the side as $25$ m. Is one of them wrong?

Check with a calculator. $20 \approx 4.4721$ , and $2 \times 5 \approx 2 \times 2.2361 = 4.4721$ . They agree to every digit the calculator shows. So both students wrote the same number in two different ways.

That raises two real questions. Why are $20$ and $25$ equal? And when the same surd can be written in more than one way, which way should we choose? The answer to the first question is one rule about square roots. The answer to the second is the skill this lecture builds: simplifying a surd.

A picture first

Before any rule, look at the garden. A square of area 20 can be cut into four smaller squares, two along each side. Each small square has area $20 \div 4 = 5$ .

A square of area 20 is four squares of area 5. Each small side is √5, so the big side is two of them: 2√5.

A square of area 5 has side $5$ . The big square's side is made of two of those small sides placed end to end. So the big side is $5 + 5$ , which is $25$ . But the big side is also $20$ , because the big square has area 20. Both describe the same length:

20 = 25

The picture worked because 20 is 4 lots of 5, and 4 is a perfect square: $2 \times 2$ . A perfect square of small squares makes a bigger square. That is the whole secret, and the next rule says it in general.

The rule behind it

The product rule for square roots

For positive numbers $a$ and $b$ : the square root of a product is the product of the square roots.

$ab = a \times b$

Why is this true? Remember what $ab$ means. It is the positive number which, multiplied by itself, gives $ab$ . So test $a \times b$ . Multiply it by itself, and rearrange the order of the multiplication:

(a \times b) \times (a \times b) = (a \times a) \times (b \times b) = a \times b

So $a \times b$ is positive, and it squares to give $ab$ . That is exactly the definition of $ab$ . The two must be equal.

Now use the rule on 20. Split 20 into a perfect square times something else, and the perfect square comes out from under the root:

20 = 4 \times 5 = 4 \times 5 = 25

The perfect square leaves the root as a whole number. The rest stays inside. That is the same thing the picture showed.

Simplest form: no square factor left inside

We now have a target. A surd is in simplest form when the number under the root has no square factor bigger than 1. A square factor is a factor that is a perfect square: 4, 9, 16, 25, 36, and so on.

So $25$ is in simplest form, because 5 has no square factor. But $20$ is not, because 20 has the square factor 4.

Why do mathematicians and examiners prefer simplest form? Because it shows the structure of the number. $72$ , $50$ and $8$ look like three unrelated numbers. In simplest form they are $62$ , $52$ and $22$ . Now you can see that all three are multiples of the same surd, $2$ . That will matter in the next lecture, when we add surds.

To reach simplest form in one step, find the largest square factor of the number under the root. Try it below with any number up to 300.

Explore — find the largest square factor

Choose a number n. The page lists its square factors, takes out the largest one, and draws the picture.

Number under the root: 72

For each number, the largest square factor comes out of the root:

Number	Square factors	Largest	Simplest form
20	4	4	$25$
50	25	25	$52$
72	4, 9, 36	36	$62$
180	4, 9, 36	36	$65$
30	none	—	$30$ , already simplest
196	4, 49, 196	196	14, a whole number

Look at 72. Its square factors are 4, 9 and 36. If you take out 4, you get $72 = 218$ . That is true, but it is not finished, because 18 still contains the square factor 9. You would have to simplify again: $218 = 2 \times 32 = 62$ . Taking out the largest square factor, 36, gets there in one step: $72 = 62$ .

Both routes reach the same answer. A smaller factor is not a mistake. It only means one more step. The rule for when you are finished never changes: no square factor left inside the root.

Stop and think

$30$ is quite a large surd. Can it be simplified?

Show the answer

No. $30 = 2 \times 3 \times 5$ . None of these primes appears twice, so 30 has no square factor bigger than 1. $30$ is already in simplest form. Whether a surd simplifies depends only on its square factors, not on how big it is. By contrast, the smaller surd $8$ does simplify, to $22$ .

When you cannot spot the square factor: use primes

For larger numbers, the largest square factor may not be obvious. Then break the number into primes, and look for pairs.

Why pairs? Because a pair of equal primes makes a perfect square. $3 \times 3 = 3^{2}$ , and $3^{2} = 3$ . So each pair inside the root comes out as a single number. Any prime without a partner stays inside.

Take 180. A factor tree gives $180 = 2 \times 2 \times 3 \times 3 \times 5$ . The pairs are $2 \times 2$ and $3 \times 3$ . Each pair sends one number outside. The 5 has no partner, so it stays inside:

180 = 2^{2} \times 3^{2} \times 5 = 2 \times 3 \times 5 = 65

Check: $6^{2} \times 5 = 36 \times 5 = 180$ . Correct. This method never misses a square factor, so it always gives simplest form in one go.

Worked examples

Worked Example 1 — Simplify √48

Square factors of 48: 4 (since $48 = 4 \times 12$ ) and 16 (since $48 = 16 \times 3$ ). 36 does not divide 48. So the largest is 16.

48 = 16 \times 3 = 43

Check that you are finished: 3 has no square factor. Check the value: $4^{2} \times 3 = 48$ .

Worked Example 2 — Simplify √200

Square factors of 200 include 4, 25 and 100. The largest is 100, because $200 = 100 \times 2$ .

200 = 100 \times 2 = 102

A useful habit: if the number ends in 00, try 100 first.

Worked Example 3 — A number already outside: simplify 3√12

$312$ means $3 \times 12$ . Simplify the root first, then multiply.

312 = 3 \times 4 \times 3 = 3 \times 23 = 63

The 2 that comes out of the root multiplies the 3 that was already outside.

Worked Example 4 — Running the rule backwards: which is bigger?

Which is larger, $43$ or $35$ ? Do not use a calculator.

The product rule also works backwards. A number outside the root can go inside if you square it first: $43 = 16 \times 3 = 48$ . In the same way, $35 = 9 \times 5 = 45$ .

Now compare $48$ with $45$ . A bigger number has a bigger square root, so $48$ is larger. So $43 > 35$ .

Check: $43 \approx 6.93$ and $35 \approx 6.71$ . Correct.

One mistake is so common that it deserves practice of its own. Find it in the solution below.

Spot the error — a hypotenuse that came out too long

A right-angled triangle has shorter sides 4 cm and 6 cm. A student finds the hypotenuse $c$ . One line is wrong. Tap the line you think it is.

A student's solution, with the error marked:

1. $c^{2} = 4^{2} + 6^{2}$ . Correct: Pythagoras' theorem.
2. $c = 4^{2} + 6^{2}$ . Correct: take the positive square root, because a length is positive.
3. $c = 4^{2} + 6^{2}$ . Wrong. A square root does not split over addition. The correct line is $c = 16 + 36 = 52$ .
4. $c = 4 + 6 = 10$ cm. Wrong, because it follows from line 3.

Correct finish: $c = 52 = 4 \times 13 = 213 \approx 7.21$ cm.

Three classic mistakes

Trap 1 — Stopping before the end

$72 = 218$ is true, but it is not simplest form, because 18 still has the square factor 9. Before you finish, always ask: does the number inside have any square factor left?

Trap 2 — Splitting a root over addition

$9 + 16$ is $25 = 5$ , but $9 + 16 = 3 + 4 = 7$ . They are not equal. The rule $ab = a b$ is for multiplication only. There is no rule that splits $a + b$ .

Trap 3 — Forgetting the number already outside

In $312$ , the root simplifies to $23$ , and the answer is $3 \times 23 = 63$ . Writing $23$ , or adding to get $53$ , loses the 3 that was already there. The number outside multiplies.

Try it yourself

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

Simplify $28$ , $45$ and $98$ .

Hint 1
For each number, test the perfect squares 4, 9, 16, 25, 36, 49 in turn. Which ones divide exactly?

Hint 2
28 = 4 × 7, 45 = 9 × 5, 98 = 49 × 2.

Full answer
$28 = 4 \times 7 = 27$ . $45 = 9 \times 5 = 35$ . $98 = 49 \times 2 = 72$ . Checks: $4 \times 7 = 28$ , $9 \times 5 = 45$ , $49 \times 2 = 98$ .
Simplify $300$ and $250$ .

Hint 1
300 ends in 00. Try 100 as the square factor.

Hint 2
For $250$ , simplify $50$ first, then multiply by the 2 outside.

Full answer
$300 = 100 \times 3 = 103$ . $250 = 2 \times 25 \times 2 = 2 \times 52 = 102$ .
Without a calculator, decide which is larger: $62$ or $53$ .

Hint 1
Move each number outside back inside its root. Square it first.

Hint 2
$62 = 36 \times 2$ . Do the same for $53$ .

Full answer
$62 = 72$ and $53 = 75$ . Since $75 > 72$ , $53$ is larger. Check: $62 \approx 8.49$ and $53 \approx 8.66$ .
Use prime factors to simplify $1200$ .

Hint 1
Start a factor tree: $1200 = 12 \times 100$ . Break both parts into primes.

Hint 2
You should find four 2s, one 3 and two 5s. Four 2s make two pairs.

Full answer
$1200 = 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5$ . The pairs are $2 \times 2$ , $2 \times 2$ and $5 \times 5$ . Each pair sends one number outside: $2 \times 2 \times 5 = 20$ . The 3 has no partner and stays inside. So $1200 = 203$ . Check: $2 0^{2} \times 3 = 400 \times 3 = 1200$ .

The bigger idea

Simplifying a surd is not tidying up for the examiner. It is finding the number's real structure. $72$ hides a fact that $62$ shows openly: this number is 6 copies of $2$ .

This is a pattern you will meet again and again in mathematics. Break a number into its prime building blocks, and its hidden properties appear. The same idea finds highest common factors, adds fractions, and later keeps the secret codes on the internet secure.

Key concepts in this lesson

The product rule $ab = a \times b$ holds for positive numbers, because $a \times b$ is positive and squares to $ab$ .
A surd is in simplest form when the number under the root has no square factor bigger than 1.
Taking out the largest square factor gives simplest form in one step, for example $72 = 36 \times 2 = 62$ .
With prime factors, each pair of equal primes comes out of the root as one number, and unpaired primes stay inside.
The rule does not work for addition, so $a + b$ is not $a + b$ .
A number outside a root can be moved inside by squaring it, which lets you compare surds such as $43 = 48$ .