Surds and Exact Calculations
Learn what surds mean before using the rules. Start with familiar square roots, then simplify, calculate and preserve exact answers.
GCSE Higher. Work through the early examples first. Conjugate rationalisation is a short, separately labelled later Higher extension.
Begin with familiar roots
You already know that √25 = 5. Some roots simplify to integers; others cannot be written as exact fractions or recurring decimals.
Those irrational roots can be kept in exact form. We call them surds.
This lesson uses positive square roots. We will simplify them, calculate with them and keep exact answers when the question asks us to.
Example Question:
Simplify √8 + √18.
The roots look different at first. Example 6 shows how square factors reveal matching root parts.
Exam Tip
Look for “exact”, “in terms of π”, “simplest surd form” or “rationalise the denominator”.
Keep roots and π throughout exact working. Use decimals only when requested or as a separate check.
Method — Simplify a square-root surd
1. Look for a square factor, preferably the largest.
2. Write the number as square factor × remaining factor.
3. Use the multiplication property to split the root.
4. Evaluate the root of the square factor.
5. Check whether the remaining root can simplify further.
Useful square factors include 4, 9, 16, 25, 36, 49 and 64.
Worked Examples
Example 1 — Recognise a surd
√25 = 5 because 5 × 5 = 25. This root simplifies to an integer.
√2 is different. Its decimal expansion never terminates or settles into a recurring block. It is irrational: it cannot be written as a fraction of two integers.
A surd is an irrational root left in exact form. This lesson focuses on square roots.
| Square root | Simplified value | Surd? |
|---|---|---|
| √4 | 2 | No |
| √9 | 3 | No |
| √16 | 4 | No |
| √2, √3, √5, √7 | Already in exact root form | Yes |
A root sign alone does not make a value a surd. Simplify or identify the root first.
Example 2 — Separate exact from approximate
√2 names the exact positive number whose square is 2.
Its decimal expansion begins 1.41421356…. To three decimal places, √2 ≈ 1.414.
The symbol ≈ means “approximately equal to”. The rounded decimal 1.414 is not exactly √2.
Another exact value is π, pronounced “pi”. It is the constant relating a circle’s circumference to its diameter.
A circumference written as 6π cm is exact. To two decimal places it is approximately 18.85 cm. The unit cm means centimetres.
If the question says “exact”, “in terms of π” or “as a surd”, keep the exact form.
Example 3 — Simplify using a square factor
Simplify √12.
12 = 4 × 3. The factor 4 is useful because √4 = 2.
For positive numbers a and b, √(ab) = √a × √b. Here a and b stand for the two factors, and ab means their product.
√12 = √(4 × 3)
= √4 × √3
= 2√3
2√3 means 2 × √3. The 2 is the coefficient, the number multiplying the root.
Check: (2√3)² = 4 × 3 = 12. Both √12 and 2√3 are positive.
The multiplication property works because (√a × √b)² = a × b, and the product of the positive roots is positive.
| Start | Useful square factor | Simplification |
|---|---|---|
| √18 | 9, since 18 = 9 × 2 | √9 × √2 = 3√2 |
| √50 | 25, since 50 = 25 × 2 | √25 × √2 = 5√2 |
Example 4 — Choose an efficient square factor
Simplify √72.
The largest square factor of 72 is 36: 72 = 36 × 2.
√72 = √36 × √2
= 6√2
A smaller square factor also works, but may need another step:
√72 = √4 × √18
= 2√18
= 2 × 3√2
= 6√2
Both routes keep the value unchanged. Finish when the number inside the root has no square factor greater than 1.
Example 5 — Collect like surds
3√2 + 5√2 means three lots of √2 plus five lots of √2.
3√2 + 5√2 = (3 + 5)√2
= 8√2
Likewise, 7√3 − 2√3 = (7 − 2)√3 = 5√3.
The root parts must match. √2 + √3 has unlike root parts and cannot be collected this way.
In particular, √2 + √3 ≠ √5. The symbol ≠ means “not equal to”. The rule for multiplying roots does not apply to adding them.
Example 6 — Simplify before collecting
Simplify √8 + √18.
√8 = √4 × √2 = 2√2
√18 = √9 × √2 = 3√2
Now the root parts match:
√8 + √18 = 2√2 + 3√2
= 5√2
Simplifying first reveals the like terms.
Example 7 — Multiply roots and coefficients
Start with √2 × √3 = √6. The numbers inside the roots multiply.
For √3 × √12:
√3 × √12 = √36
= 6
Now simplify 2√3 × 4√5.
Multiply the coefficients: 2 × 4 = 8.
Multiply the roots: √3 × √5 = √15.
Answer: 8√15. Fifteen has no square factor greater than 1, so this is fully simplified.
Divide surds
For a ≥ 0 and b > 0, √a ÷ √b = √(a ÷ b). Divide the coefficients separately and simplify the root. The denominator must not be zero.
Worked example: 6√18 ÷ (3√2) = (6 ÷ 3) × √(18 ÷ 2) = 2√9 = 6. Check by multiplying: 6 × 3√2 = 18√2, which also equals 6√18.
Quick practice — Simplify 4√15 ÷ (2√3).
(4 ÷ 2) × √(15 ÷ 3) = 2√5.
Example 8 — Square the whole expression
A square root undoes a square: (√5)² = √5 × √5 = 5.
Now find (3√2)². The brackets mean square the whole product.
(3√2)² = (3√2) × (3√2)
= (3 × 3) × (√2 × √2)
= 9 × 2
= 18
Remember to square the coefficient as well as the root.
Example 9 — Expand brackets carefully
Expand √2(3 + √2). Multiply √2 by each term inside:
√2 × 3 + √2 × √2
= 3√2 + 2
Now expand (2 + √3)(1 + √3). Each term in the first bracket multiplies each term in the second.
| Product | Result |
|---|---|
| 2 × 1 | 2 |
| 2 × √3 | 2√3 |
| √3 × 1 | √3 |
| √3 × √3 | 3 |
Combine the four results:
2 + 2√3 + √3 + 3
= 5 + 3√3
Example 10 — Rationalise a single-surd denominator
Rationalising rewrites a fraction with a rational denominator. Rational means expressible as a fraction of integers; the new denominators here are whole numbers.
This can make exact answers easier to compare or combine, and a question may request it. The original fraction is already exact.
Rewrite . Multiply both numerator and denominator by √3.
That multiplies the fraction by , which equals 1, so the value stays the same.
=
=
The denominator is rational because √3 × √3 = 3.
Similarly, = = .
The root has not simply moved upstairs. We multiplied the top and bottom by the same non-zero value.
Example 11 — Keep exact values in measurement answers
A circle has radius 4 cm. Give its exact area.
Area = πr², where r is the radius and r² means r × r.
Area = π × 4²
= 16π cm²
The unit cm² means square centimetres. Leave π in the answer because exact area was requested.
Now a square has area 18 cm². Find its exact side length.
Side length = √18
= 3√2 cm
Check: (3√2)² = 18. A side length uses cm, not cm².
Example 12 — Later Higher extension: a conjugate
Rationalise . Multiplying by √3 alone would leave another surd term in the denominator.
The conjugate of 2 + √3 is 2 − √3. It has the same two terms with the sign between them reversed.
Multiply numerator and denominator by 2 − √3. This is non-zero, so the multiplier equals 1.
First expand the denominator:
(2 + √3)(2 − √3)
= 4 − 2√3 + 2√3 − 3
= 4 − 3
= 1
The middle terms cancel. This is the difference of two squares: 2² − (√3)².
=
=
= 2 − √3
Check: (2 − √3)(2 + √3) = 1, so the answer multiplied by the original denominator returns the numerator.
Key ideas to remember
Simplify before collecting like surds.
For multiplication, multiply coefficients and root parts separately, then simplify.
Squaring a product means squaring every factor: (3√2)² = 9 × 2.
Rationalising changes the form of a fraction, not its value.
π is an exact irrational constant, but it is not itself a surd. A surd is an irrational root.
Calculator and answer checks
For a simplified positive root, square the result: (5√2)² = 50 checks √50 = 5√2.
Compare numerical values as a check: 2√3 ≈ 3.464 to three decimal places. Keep 2√3 as the exact answer.
Use your calculator’s root key for the whole intended expression and keep brackets around sums in a denominator.
For a rationalised fraction, multiply the answer by the original denominator. It should return the numerator.
Make sure terms really have the same simplified root before collecting. Check measurement units and whether the question asked for an approximation.
Think Like a Mathematician
Why can √8 + √18 be collected, but √2 + √3 cannot?
√8 contains two lots of √2. √18 contains three lots of √2.
We are adding the same-sized quantity five times, so the total is 5√2.
√2 and √3 are different quantities. There is no shared root part to collect.
As a check on the tempting wrong answer, (√2 + √3)² = 5 + 2√6, whereas (√5)² = 5. These are not the same.
Common mistakes to avoid
Calling every root a surd: √16 equals the integer 4.
Splitting a product as a sum: √12 = √4 × √3, not √4 + √3.
Adding inside roots: √2 + √3 is not √5.
Stopping early: 2√18 still simplifies to 6√2.
Losing coefficients: keep the number multiplying a root when you simplify it.
Not finishing a root product: √3 × √3 = √9 = 3.
Rounding exact working: leave roots or π in an exact answer.
Changing only the denominator: rationalise by multiplying both top and bottom by the same non-zero value.
Try it yourself
Show exact working. Questions 12 and 13 are the harder Higher bracket and conjugate steps; Question 14 checks simplification with an existing coefficient.
1. Which are surds: √3, √16, √20 and √49?
√16 = 4 and √49 = 7, so neither is a surd.
√3 remains irrational. √20 = √4 × √5 = 2√5, still irrational.
The surds are √3 and √20.
2. A calculator gives √7 ≈ 2.646. Which form is exact? Also give an exact circumference when it is stated as 10π cm.
√7 is exact; 2.646 is rounded to three decimal places.
The exact circumference is already 10π cm. Keep π rather than replacing it with a rounded decimal.
3. Simplify √20 and √45.
20 = 4 × 5, so √20 = √4 × √5 = 2√5.
45 = 9 × 5, so √45 = √9 × √5 = 3√5.
Five has no square factor greater than 1.
4. Simplify √98 using its largest square factor.
98 = 49 × 2, and 49 is a square number.
√98 = √49 × √2 = 7√2.
Check: (7√2)² = 49 × 2 = 98.
5. Simplify 6√3 − 2√3 + √3.
Every term has the same √3. The last term has coefficient 1.
6 − 2 + 1 = 5.
Answer: 5√3.
6. Simplify √12 + √27.
√12 = √4 × √3 = 2√3.
√27 = √9 × √3 = 3√3.
2√3 + 3√3 = 5√3.
7. Simplify 3√2 × 2√6.
Coefficients: 3 × 2 = 6.
Roots: √2 × √6 = √12.
6√12 = 6 × 2√3 = 12√3.
8. Work out (2√5)².
(2√5)² = (2 × 2) × (√5 × √5).
= 4 × 5
= 20.
9. Expand √3(2 + √3).
Multiply √3 by both terms:
√3 × 2 + √3 × √3
= 2√3 + 3.
10. Rationalise .
Multiply top and bottom by √7:
=
= .
The multiplier equals 1. Check: × √7 = 2.
11. A circle has radius 3 cm. Give its exact area. A square has area 50 cm²; give its exact side length.
Circle area = π × 3² = 9π cm².
Square side = √50 = √25 × √2 = 5√2 cm.
Check: (5√2)² = 50. Keep both answers exact.
12. Harder Higher: expand and simplify (3 + √2)(2 − √2).
Multiply all four pairs:
6 − 3√2 + 2√2 − 2
Combine whole numbers: 6 − 2 = 4.
Combine root terms: −3√2 + 2√2 = −√2.
Answer: 4 − √2.
13. Later Higher extension: rationalise .
Use the conjugate 3 − √5. Multiply top and bottom by it.
Denominator: (3 + √5)(3 − √5)
= 9 − 3√5 + 3√5 − 5
= 4
Answer: .
Check: multiplying this answer by 3 + √5 gives (9 − 5) ÷ 4 = 1.
14. Simplify fully: 2√18 − √8.
√18 = 3√2, so 2√18 = 6√2.
√8 = 2√2.
6√2 − 2√2 = 4√2.
Keep the original coefficient 2 when simplifying the first term.
Frequently Asked Questions
Is √9 a surd?
No. It simplifies to 3, which is rational.
Is √12 still a surd if it simplifies?
Yes. It simplifies to 2√3, which is still irrational. Simplifying a surd does not always remove the root.
Do I have to use the largest square factor?
No. A smaller square factor works too, but may require another simplification step.
Can I collect √2 and √3?
No. Their simplified root parts differ. Keep the sum as √2 + √3.
Is a fraction with a surd denominator wrong?
No; it can already be exact. Rationalising gives an equivalent form with a rational denominator, as a question may request.
When should I use a decimal?
When a question requests an approximation or stated accuracy. Use decimal values separately to check exact working.
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