Square Root Calculator - Calculate √x, Cube Roots & Nth Roots

What is a Square Root?

A square root of a number x is a value y such that y² = x. For example, √9 = 3 because 3² = 9. The symbol √ is called the radical sign. Every positive number has two square roots: one positive (principal square root) and one negative.

Types of Roots

Square Root (√): The number that multiplied by itself gives the original number
Cube Root (∛): The number that multiplied by itself three times gives the original
Nth Root: General case where the result raised to power n equals the original

Perfect Squares

A perfect square is a number whose square root is an integer. The first 15 perfect squares are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Perfect Cubes

A perfect cube is a number whose cube root is an integer. The first 10 perfect cubes are:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Properties of Square Roots

√(a × b) = √a × √b
√(a / b) = √a / √b
(√a)² = a
√a² = |a| (absolute value)

Simplifying Radicals

A radical is in simplified form when the number under the radical sign has no perfect square factors (other than 1). For example, √12 = √(4×3) = 2√3.

Frequently Asked Questions

What is the square root of a negative number?

Even roots (square root, fourth root, etc.) of negative numbers are not defined in the real number system. They exist as imaginary numbers in the complex plane. Odd roots (cube root, fifth root) of negative numbers are real and negative.

What is a perfect square?

A perfect square is a number whose square root is a whole number. The first few are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225. This calculator automatically detects perfect squares, cubes, and higher powers.

How do I simplify a square root?

Find the largest perfect square factor of the number and split the radical. For example, √12 = √(4×3) = 2√3. This calculator shows the simplified radical form automatically when applicable.