Simplify square roots, cube roots, and nth roots to their simplest radical form (a√b). Shows all perfect square factors, step-by-step working, and decimal approximation for UK GCSE surds and US Algebra radicals.
Simplifying square roots — expressing them in the form a√b where b has no perfect square factors — is a fundamental skill in algebra. In the United Kingdom, simplified square roots are called surds and are a required topic in GCSE Mathematics. In the United States, they appear as simplified radical expressions in Algebra 1 and Algebra 2. This free square root simplifier handles square roots, cube roots, and nth roots, with complete step-by-step working.
A radical expression √n is in simplest form when the radicand n contains no perfect square factors other than 1. In other words, you cannot pull any whole number out from under the radical sign.
For example: √72 is not in simplest form because 72 = 36 × 2, and 36 = 6² is a perfect square. Pulling out √36 = 6 gives: √72 = √(36 × 2) = 6√2. Since 2 has no perfect square factors, 6√2 is the simplified form.
| n | n² | n | n² |
|---|---|---|---|
| 1 | 1 | 8 | 64 |
| 2 | 4 | 9 | 81 |
| 3 | 9 | 10 | 100 |
| 4 | 16 | 11 | 121 |
| 5 | 25 | 12 | 144 |
| 6 | 36 | 13 | 169 |
| 7 | 49 | 15 | 225 |
The key rule used is: √(a × b) = √a × √b for non-negative a and b.
Like terms with radicals can be added or subtracted — but only if they have the same radicand (the number under the radical). For example: 3√2 + 5√2 = 8√2. But √2 + √3 cannot be simplified further.
To add unlike radicals, first simplify each one: √12 + √27 = 2√3 + 3√3 = 5√3.
To multiply radicals: √a × √b = √(ab). For example: √6 × √10 = √60 = √(4 × 15) = 2√15.
Mixed radicals multiply as: (a√b) × (c√d) = ac√(bd). For example: 3√2 × 4√5 = 12√10.
Expressions with radicals in the denominator are considered "not simplified." To rationalise: multiply numerator and denominator by the radical. For example: 1/√3 = (1/√3) × (√3/√3) = √3/3.
For conjugate expressions like 1/(√5 + 2): multiply by (√5 − 2)/(√5 − 2) = (√5 − 2) / (5 − 4) = √5 − 2. This is the conjugate method.
In the UK, rationalising the denominator is a required GCSE Higher topic. In the US, it appears in Algebra 2 and Pre-Calculus.
The same principle applies to cube roots: ∛n is simplified when n has no perfect cube factors. For example: ∛54 = ∛(27 × 2) = 3∛2.
For an nth root: ⁿ√n is simplified when the radicand has no perfect nth power factors. The rule is: ⁿ√(a^n × b) = a × ⁿ√b.
The square root of a negative number is imaginary: √(−1) = i, where i is the imaginary unit. For any positive n: √(−n) = i√n. For example: √(−12) = i√12 = 2i√3.
Imaginary numbers appear in Algebra 2 (US) and A-Level Mathematics (UK). The complex number system (a + bi) extends the real numbers to include square roots of negatives.
In England, surds are explicitly named and taught in GCSE Mathematics (both Foundation and Higher tiers). Students are expected to simplify surds, rationalise denominators, and recognise that surds are exact irrational numbers. Expressing answers as surds (rather than decimals) is often required in exam questions where "exact form" is specified. AQA, Edexcel, and OCR all include surd questions on GCSE papers.
A Level Mathematics extends this to surd arithmetic in proof, rationalising complex denominators, and simplifying expressions involving surds in trigonometry (sin 30° = 1/2, cos 30° = √3/2, etc.).
In the United States, radical simplification is taught in Algebra 1 (grades 8–9) and extended in Algebra 2. The Common Core Standards require students to simplify square roots and understand the product property of radicals. Simplified radical form is the standard expected answer on standardised tests including the SAT and ACT.
Find the largest perfect square factor of the radicand. Write it as a product, then take the square root of the perfect square part outside. For √72: largest perfect square factor = 36, so √72 = √(36×2) = 6√2.
A surd is an irrational number that can be expressed as the root of a non-perfect-square integer. Examples: √2, √3, √5, √7. They cannot be expressed as exact fractions. Surds are taught at GCSE and A-Level in England, Wales, and Northern Ireland.
√4 = 2 exactly (a rational number — 4 is a perfect square). √5 ≈ 2.236... and cannot be expressed exactly as a fraction — it is an irrational surd.
You can only add radicals with the same radicand (like terms): 3√2 + 5√2 = 8√2. First simplify: √12 + √27 = 2√3 + 3√3 = 5√3. Different radicands like √2 + √3 cannot be combined.
It means removing any radical from the denominator. For 1/√3: multiply top and bottom by √3 to get √3/3. For 1/(√5+2): multiply by the conjugate (√5−2) to get √5−2.
Not as a real number. √(−1) = i (the imaginary unit). For any negative radicand: √(−n) = i√n. This is part of the complex number system taught in Algebra 2 (US) and A-Level Maths (UK).
Find the largest perfect cube factor. For ∛54: 54 = 27×2, and 27 = 3³, so ∛54 = 3∛2. The process is the same as square roots but using perfect cubes instead of perfect squares.
Yes. By the product property: √2 × √2 = √(2×2) = √4 = 2. More generally, √n × √n = n for any non-negative n. This is used in rationalising denominators.
Results are for educational purposes. All simplifications use standard algebraic rules. Decimal approximations are rounded to 10 significant figures.