To learn Surds quickly and correctly, start with one clear idea: a surd is a root expression that cannot be simplified to a rational number. For example, √2 is an irrational number, but √9 = 3 is not a surd in simplified form. Surds help us keep exact answers instead of rounding too early. Once you can recognise perfect-square factors, combine like terms and understand why brackets and denominators matter, surd questions become familiar algebra. These eduKateSG Mathematics Quick Notes show the key methods through easy examples and quick accuracy checks.
- Recognise a surd: √2 and √5 cannot be written exactly as ordinary fractions.
- Simplify: take a perfect-square factor outside a square root, such as √50 = 5√2.
- Add and subtract: combine only like surd terms after simplifying.
- Multiply: use ordinary multiplication and valid root laws.
- Rationalise: remove a surd from the denominator using an equivalent expression.
- Check: preserve exact form and use a decimal approximation only as a reasonableness test.
First, learn the difference between an exact form and a rounded decimal. The value √2 is approximately 1.414, but no short decimal is exactly √2. If a question asks for an exact answer, leaving √2 is often more correct than writing 1.41. For example, the diagonal of a square with side 1 cm is √2 cm by Pythagoras’ theorem. You can use a calculator to check that the answer is near 1.414 cm, but you should retain √2 when the question specifies exact form.
Second, simplify a square root by finding its largest convenient perfect-square factor. Consider √72. Since 72 = 36 × 2, write √72 = √36 × √2 = 6√2. This step works for non-negative real factors because √(ab) = √a × √b when a and b are non-negative. The aim is to bring square factors such as 4, 9, 16, 25 or 36 outside the root. You can check 6√2 by squaring it: (6√2)² = 36 × 2 = 72.
Third, treat surds like algebraic terms. The expression 3√5 + 2√5 has two quantities of the same kind, so it equals 5√5. But √2 + √3 cannot be combined into √5; the two roots are different, just as x and y are different variables. Before adding surds, simplify each one. For example, √12 + √27 = 2√3 + 3√3 = 5√3. Recognising the same remaining root is what lets you combine terms.
Worked example: simplify √98
Find a perfect-square factor of 98. Since 98 = 49 × 2, √98 = √49 × √2 = 7√2. Check with a calculator: √98 is approximately 9.899, and 7 × √2 is also approximately 9.899. The exact answer 7√2 gives the full mathematical value, while the decimal confirms its approximate size.
Worked example: add and subtract unlike-looking surds
Simplify 2√18 + √8 − √50. First simplify each root: √18 = 3√2, √8 = 2√2 and √50 = 5√2. Then 2√18 = 6√2, so the whole expression becomes 6√2 + 2√2 − 5√2 = 3√2. Do not combine the original numbers 18, 8 and 50 under a single square root. The valid step is to simplify the terms first, then collect matching √2 parts.
Worked example: multiply surds
Calculate √6 × √24. Both quantities under the roots are non-negative, so their product is √(6 × 24) = √144 = 12. Alternatively, simplify √24 = 2√6, so √6 × 2√6 = 2 × 6 = 12. The two routes agree. Remember that when a square root is multiplied by itself, √a × √a = a for a ≥ 0; you do not leave the answer as another √a.
Expand brackets that contain surds
Expand (√3 + 2)(√3 − 2). This is a difference of squares: (a + b)(a − b) = a² − b². Therefore the answer is (√3)² − 2² = 3 − 4 = −1. You can also distribute every term and watch the middle terms cancel. The familiar algebra laws still apply; a surd is a number, so you multiply it just as you would any other exact quantity.
Another bracket example with like surds
Expand (√5 + √2)². This means (√5 + √2)(√5 + √2). The four terms are 5, √10, √10 and 2. Combine the middle terms: 7 + 2√10. A common error is to square each term and forget the two cross-products, giving 5 + 2 = 7. The correct expansion of (a + b)² is a² + 2ab + b², not just a² + b².
What does rationalising a denominator mean?
A denominator such as √2 can often be rewritten without a square root while keeping the fraction’s value unchanged. For example, 5/√2 can be multiplied by √2/√2, which equals 1. The result is 5√2/2, so 5/√2 = 5√2/2. The new denominator is rational. Nothing about the value has changed; you have simply rewritten the exact number in a form that is often preferred in school Mathematics.
Worked example: rationalise with a conjugate
Simplify 1/(√3 + 1). Multiply the numerator and denominator by the conjugate √3 − 1. The denominator becomes (√3 + 1)(√3 − 1) = 3 − 1 = 2. Therefore the result is (√3 − 1)/2. A conjugate changes the sign between two terms so that the middle products cancel. This technique is useful when the denominator contains a sum or difference involving a surd.
Know which square-root rules are valid
- Valid for a, b ≥ 0: √(ab) = √a × √b.
- Valid for a ≥ 0, b > 0: √(a/b) = √a/√b.
- Not generally valid: √(a + b) = √a + √b.
- Not generally valid: √(a − b) = √a − √b.
- Important for real x: √(x²) = |x|.
- When an equation says x² = a > 0: there are two real solutions, x = ±√a; the symbol √a itself means the non-negative root.
Why √9 + √16 is not √25
Try the two calculations. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. The answers differ, which proves that moving addition inside or outside a radical is not a valid general rule. If you find yourself wanting to merge roots, ask whether you are multiplying or adding them. Product rules may apply; sum rules usually do not.
When should an answer be left exact?
If a Mathematics question asks for an exact value, write answers such as 3√2, √13 or (√3 − 1)/2 rather than prematurely replacing them with rounded decimals. If the question asks for a measurement to three significant figures, perform the algebra exactly first, then evaluate and round at the end. Keeping the surd protects accuracy through multi-step calculations. An exact answer can always be approximated later, but an early rounded answer may introduce avoidable error.
Common surd mistakes
- Adding different roots: √2 + √3 cannot be simplified to √5.
- Failing to simplify first: √8 + √18 becomes 2√2 + 3√2 = 5√2.
- Forgetting cross-products in brackets: (√5 + √2)² includes 2√10.
- Using a root-of-sum rule: √(a + b) is not generally √a + √b.
- Dropping a restriction: real square roots require non-negative radicands, and denominators must be non-zero.
- Rounding each root before calculating: keep exact surds until the final requested precision.
Mini-test: practise without copying
- Simplify √75.
- Simplify √12 + √27.
- Calculate √8 × √2.
- Expand (√7 + 1)(√7 − 1).
- Rationalise 3/√5.
Answers and explanations
1. √75 = √(25 × 3) = 5√3. 2. 2√3 + 3√3 = 5√3. 3. √16 = 4. 4. Difference of squares: 7 − 1 = 6. 5. Multiply by √5/√5 to obtain 3√5/5. Each answer can be checked by estimating its decimal value.
Your ten-minute surd learning routine
- Minutes 1–2: identify perfect squares inside five small roots.
- Minutes 3–4: simplify two surds and combine like terms.
- Minutes 5–6: multiply a pair of surds and expand a short bracket.
- Minutes 7–8: rationalise a simple denominator.
- Minutes 9–10: solve one mixed question, keep an exact answer and check it with a decimal estimate.
Surds are not complicated new objects. They are exact numbers that follow familiar arithmetic and algebra rules. Once you know what can be factored, combined or cancelled, the steps become predictable.
Learn the next connected idea
Continue with Square Roots and Cube Roots · Indices · Algebraic Expansion and Factorisation · Quadratic Equations. Check your understanding by answering a changed question from memory tomorrow.
Read the broader teaching pathway in Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. The original reference article remains unchanged.