Surds Calculator
Simplify, add, subtract, multiply, and divide surds (square roots) instantly. Get exact mathematical forms with clear, step-by-step breakdowns.
Calculate & Simplify Surds
Select your operation, enter the values, and instantly see the simplified exact surd form along with a step-by-step mathematical working.
Surds Solver
Get exact answers with step-by-step working.
Surds Rules
Key principles for working with roots
Simplifying
Find the largest perfect square factor of the number under the root. Split the root and take the square root of the perfect square outside.
Adding / Subtracting
You can only combine ‘like surds’ (those with the same number under the root). Treat the surd part like a variable (e.g., 3√5 + 2√5 = 5√5).
Multiplying
Multiply the coefficients (numbers outside) together, and multiply the radicands (numbers inside) together. Then simplify the result.
Rationalising
To divide by a surd, multiply the numerator and denominator by that surd to remove the root from the bottom of the fraction.
Common Surd Simplifications
Frequently encountered surds in mathematics and their simplified exact forms.
| Original Surd | Simplified Exact Form | Decimal Approximation |
|---|---|---|
| √8 | 2√2 | ~2.828 |
| √12 | 2√3 | ~3.464 |
| √18 | 3√2 | ~4.243 |
| √20 | 2√5 | ~4.472 |
| √27 | 3√3 | ~5.196 |
| √32 | 4√2 | ~5.657 |
| √45 | 3√5 | ~6.708 |
| √50 | 5√2 | ~7.071 |
| √72 | 6√2 | ~8.485 |
| √75 | 5√3 | ~8.660 |
Surds FAQ
Answers to the most frequently asked questions about simplifying and calculating with surds in mathematics.
A surd is a number that cannot be simplified to remove a square root (or cube root, etc.). For example, √2 is a surd because it cannot be simplified to a whole number, whereas √4 is not a surd because it simplifies exactly to 2.
To simplify a surd, find the largest perfect square that is a factor of the number under the root. For example, to simplify √72, the largest perfect square factor is 36. Therefore, √72 = √(36 × 2) = 6√2.
You can only directly add or subtract surds if they have the same number under the root (known as ‘like surds’). For example, 2√3 + 5√3 = 7√3. However, √2 + √3 cannot be combined. Sometimes, simplifying the surds first reveals that they are like surds (e.g., √2 + √8 = √2 + 2√2 = 3√2).
To rationalise a denominator containing a single surd, multiply both the numerator and the denominator by that surd. For example, to rationalise 1/√2, multiply the top and bottom by √2 to get √2/2. This removes the surd from the denominator.