- Enter any positive number. For square roots of negative numbers, the result is imaginary (not calculated here). Decimals and large numbers are fully supported.
- Select the root type. Square root (2nd root, n=2) is the default. Cube root (n=3) finds what number cubed equals the input. Change n to any value for any nth root.
- Read the results. The calculator shows the decimal result, the simplified radical form (e.g., √50 = 5√2), and the step-by-step factorization. The inverse (squaring the result) is also shown to verify accuracy.
Square Root Calculator
Calculate square roots, cube roots, and nth roots with step-by-step solutions.
Square Root
12
Simplified: 12
Perfect square
| Exact form | √144 |
| Simplified radical | 12 |
| Decimal | 12. |
Perfect Squares Reference
| Number | Square Root |
|---|---|
| 81 | 9 |
| 100 | 10 |
| 121 | 11 |
| 144 | 12 |
| 169 | 13 |
| 196 | 14 |
| 225 | 15 |
How to Use the Square Root Calculator
Square Root and Radical Formulas
Square root: √x = x^(1/2) Cube root: ∛x = x^(1/3) Nth root: ⁿ√x = x^(1/n) Product rule: √(a × b) = √a × √b Quotient rule: √(a/b) = √a / √b Power rule: √(aⁿ) = a^(n/2) Simplifying: √72 = √(36 × 2) = 6√2 √200 = √(100 × 2) = 10√2 √48 = √(16 × 3) = 4√3
Perfect squares (their square roots are whole numbers):
| Number | Square Root | Number | Square Root |
|---|---|---|---|
| 1 | 1 | 64 | 8 |
| 4 | 2 | 81 | 9 |
| 9 | 3 | 100 | 10 |
| 16 | 4 | 144 | 12 |
| 25 | 5 | 169 | 13 |
| 36 | 6 | 225 | 15 |
| 49 | 7 | 400 | 20 |
Frequently Asked Questions
The square root of a number x is the value that, when multiplied by itself, gives x. √25 = 5 because 5 × 5 = 25. Every positive number has two square roots: a positive one (5) and a negative one (-5), since (-5) × (-5) = 25 as well. By convention, the square root symbol √ refers to the positive root. Zero has one square root: zero itself.
A perfect square is a positive integer whose square root is also a whole integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... Perfect squares arise by multiplying any integer by itself. Non-perfect squares (like 2, 3, 5, 7, 8) have irrational square roots: decimals that never end or repeat. √2 = 1.41421356..., √3 = 1.73205080..., √5 = 2.23606797...
Factor out the largest perfect square from inside the radical. √72 = √(36 × 2) = √36 × √2 = 6√2. √200 = √(100 × 2) = 10√2. To find the largest perfect square factor: list perfect squares (4, 9, 16, 25, 36...) and find the largest one that divides your number evenly. For √108: 4 divides 108 (27), 9 divides 108 (12), 36 divides 108 (3). The largest is 36, so √108 = 6√3.
√2 ≈ 1.41421356... It is irrational, meaning it cannot be expressed as a fraction. This was one of the first numbers proven irrational by the ancient Greeks (attributed to Hippasus, around 500 BC). It equals the diagonal of a unit square (sides = 1) by the Pythagorean theorem: √(1² + 1²) = √2. It appears in the √2 aspect ratio used for international paper sizes (A4, A3, etc.) so sheets scale perfectly when halved.
The cube root of x (∛x) is the number that cubed gives x. ∛8 = 2 because 2³ = 8. ∛27 = 3, ∛125 = 5. Unlike square roots, cube roots of negative numbers are real (not imaginary): ∛(-8) = -2 because (-2)³ = -8. The cube root is written as x^(1/3) and represents the side length of a cube with the given volume. A cube with volume 64 ft³ has side length ∛64 = 4 ft.
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