Square Roots and Simplifying Radicals Calculator

A square root is the value that, when multiplied by itself, gives the original number. Simplifying radicals means expressing a square root in its simplest form by factoring out perfect squares from the radicand.

What is a Square Root?

The square root of a number x is a value y such that y² = x. For example, the square root of 16 is 4 because 4 × 4 = 16. Square roots can be positive or negative, but the principal (or positive) square root is typically used in mathematical contexts.

For a non-negative real number x, the principal square root is denoted as √x.

Square roots of non-perfect squares are irrational numbers that cannot be expressed as simple fractions. These are called radicals and are written with the radical symbol (√).

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring out perfect squares from the radicand (the number under the radical symbol).

Steps to Simplify Radicals

Factor the radicand into perfect squares and other factors.
Separate the square root of the perfect square from the other factors.
Simplify the square root of the perfect square.

Example: Simplify √72

1. Factor 72: 72 = 36 × 2

2. √72 = √(36 × 2) = √36 × √2 = 6√2

Rules for Simplifying Radicals

√(a × b) = √a × √b
√(a/b) = √a / √b
√(a² × b) = a√b (if a is positive)

Using the Calculator

Our calculator can compute square roots and simplify radicals. Enter the radicand in the input field and click "Calculate" to see the result.

Assumptions:

Only principal (positive) square roots are calculated.
The radicand must be a non-negative number.
Results are displayed with up to 6 decimal places.

Worked Examples

Example 1: Simple Square Root

Calculate √25

Solution: 25 is a perfect square (5 × 5), so √25 = 5.

Example 2: Simplifying Radicals

Simplify √80

Solution: Factor 80 = 16 × 5, so √80 = √(16 × 5) = √16 × √5 = 4√5.

FAQ

What is the difference between a square root and a square?: The square of a number is the result of multiplying the number by itself (e.g., 5² = 25). The square root of a number is a value that, when multiplied by itself, gives the original number (e.g., √25 = 5).
Can I simplify √(a + b)?: No, the sum inside a square root cannot be simplified further unless a and b are perfect squares that can be factored out.
What is the square root of a negative number?: The square root of a negative number is not a real number. It is an imaginary number, represented as √(-1) = i, where i is the imaginary unit.