What is a square root and how is it calculated?

The square root of a number x is a value that, when multiplied by itself, equals x. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots (positive and negative), though the √ symbol typically denotes the principal (positive) square root.

How do cube roots differ from square roots?

A cube root (∛x) is a number that, when multiplied by itself three times, equals x. For example, ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots of negative numbers exist (e.g., ∛(-8) = -2), since multiplying three negative numbers gives a negative result.

Can you find square roots of negative numbers?

In the real number system, square roots of negative numbers don't exist (no real number squared gives a negative result). However, in the complex number system, √(-1) = i (the imaginary unit), and √(-4) = 2i. Complex numbers extend mathematics to handle these cases.

How do you simplify square roots?

To simplify a square root, factor out perfect squares. For example, √72 = √(36 × 2) = √36 × √2 = 6√2. Find the largest perfect square factor, take its square root outside the radical, and leave the rest inside. This form is often required in mathematical answers.

What's the relationship between square roots and exponents?

Square roots are equivalent to the exponent 1/2: √x = x^(1/2). Similarly, cube roots equal x^(1/3), and nth roots equal x^(1/n). This connection allows you to use exponent rules to manipulate roots algebraically.

Math

Square Root

Calculate square roots and cube roots.

How to Use This Calculator

Choose root type

Select whether you want to calculate a square root (√), cube root (∛), or other nth root. Each root type has different mathematical properties and applications.

Enter the number

Input the value for which you want to find the root. For square roots, enter a non-negative number unless working with complex numbers. Cube roots and odd roots accept negative numbers.

View the result and simplification

The calculator displays both the decimal approximation and the simplified radical form (if applicable). For example, √50 shows both 7.071... and 5√2, helping you understand both practical and exact mathematical forms.