A square root of a number x is a value that, when multiplied by itself, gives x. For example, √9 = 3 because 3 × 3 = 9. The symbol √ is called the radical sign. Every positive number has two square roots: one positive (principal root) and one negative. We usually refer to the positive root.

A square root (√) is a number that when multiplied by itself gives the original number (2 factors). A cube root (∛) is a number that when multiplied by itself three times gives the original number (3 factors). For example: √16 = 4 (because 4×4=16), but ∛64 = 4 (because 4×4×4=64).

An nth root is a generalization of square and cube roots. The nth root of x (written as ⁿ√x) is a number that, when raised to the power n, equals x. Mathematically, ⁿ√x = x^(1/n). For example, the 4th root of 16 is 2 because 2⁴ = 16.

A perfect square is an integer that is the square of another integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144... These are perfect squares because √1=1, √4=2, √9=3, etc. Our calculator automatically detects and highlights perfect squares.

To simplify a radical, factor out perfect squares. For example: √72 = √(36 × 2) = √36 × √2 = 6√2. Our calculator automatically shows simplified forms when possible, extracting the largest perfect square factor.

In real numbers, the square root of a negative number is undefined because no real number multiplied by itself gives a negative result. However, in complex numbers, √(-1) = i (imaginary unit). Our calculator works with real numbers and shows 'Undefined' for √negative numbers.

√2 ≈ 1.41421356... It's an irrational number, meaning it cannot be expressed as a fraction and its decimal expansion goes on forever without repeating. It's also known as Pythagoras' constant and appears frequently in geometry, especially in right triangles.

This calculator uses JavaScript's IEEE 754 double-precision floating-point arithmetic, providing about 15-17 significant decimal digits of precision. You can adjust the display precision from 0 to 12 decimal places using the slider.

Roots are the inverse of exponents. The nth root of x equals x raised to the power of 1/n. So: √x = x^(1/2), ∛x = x^(1/3), ⁴√x = x^(1/4), and so on. This relationship is fundamental in algebra and calculus.

Batch mode allows you to calculate roots for multiple numbers at once. Enter each number on a new line (or upload a file), and the calculator will show √, ∛, and nth root for all of them in a table. Results can be exported as CSV.

The Reference Table shows 30+ commonly used numbers from 1 to 1000, including all perfect squares up to 400, perfect cubes, and useful values like 2, 3, 5, 10, 100. Each entry shows √, ∛, squared, cubed, and whether it's a perfect square.

√2 is the diagonal of a unit square (a square with side length 1). It was one of the first numbers proven to be irrational by the ancient Greeks. It appears in the Pythagorean theorem, 45-45-90 triangles, and many engineering calculations.

Unlike square roots, cube roots of negative numbers are defined in real numbers. For example, ∛(-8) = -2 because (-2) × (-2) × (-2) = -8. Our calculator correctly handles negative numbers for cube roots and odd nth roots.

The golden ratio φ ≈ 1.618 is calculated as (1 + √5) / 2. It appears in art, architecture, nature (spiral shells, flower petals), and has unique mathematical properties like φ² = φ + 1.

Yes! Click 'Export as CSV' to download your results. Single mode exports the current number's roots and properties. Batch mode exports all processed numbers with their roots. Reference Table exports the entire lookup table.