Square Root Calculator - Find Square Roots with Step-by-Step Solutions

Square Root Calculator

Calculate square roots of positive, negative, or zero values. Learn manual calculation methods, understand real vs complex roots, and explore mathematical applications.

100% FreeStep-by-step SolutionsEducational Focus

Square Root Calculator

Enter any number to calculate its square root

Number *

Can be positive, negative, or zero

Show complex number results for negative inputs

Try These Examples

Click on any example to automatically calculate

Example

\sqrt{25}

Perfect square example

Example

\sqrt{12}

Non-perfect square

Example

\sqrt{0}

Square root of zero

Example

\sqrt{-4}

Negative number (complex result)

Example

\sqrt{144}

Larger perfect square

Example

\sqrt{2}

Irrational square root

What is a Square Root?

A square root of a number x is a value that, when multiplied by itself, gives x. In mathematical notation: if $y^2 = x$ , then y is a square root of x.

Key Concepts:

Principal Root: The non-negative square root ( $\sqrt{x}$ )
Positive Numbers: Have two real square roots (positive and negative)
Zero: Has exactly one square root: 0
Negative Numbers: No real square roots (complex roots exist)

Example: $\sqrt{9} = 3$ because $3^2 = 9$
But 9 also has a negative square root: -3, because $(-3)^2 = 9$

How to Calculate Square Roots Manually?

Method 1: Prime Factorization

Factor the number into prime factors
Group identical factors in pairs
Take one number from each pair outside the radical
Multiply the numbers outside

Method 2: Long Division Method

Group digits in pairs from right to left
Find the largest square ≤ first group
Subtract and bring down next group
Double the quotient and continue

Method 3: Estimation & Refinement

Find perfect squares above and below
Make an initial estimate
Use Newton's method: $x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$

Applications of Square Roots

Geometry

Finding side lengths of squares
Pythagorean theorem calculations
Distance between points
Areas and perimeters

Physics & Engineering

Velocity calculations (v = $\sqrt{2gh}$ )
RMS values in electricity
Standard deviation in statistics
Wave equations and oscillations

Algebra

Solving quadratic equations
Simplifying radical expressions
Finding domain of functions
Complex number calculations

Perfect Squares Reference Table (1-20)

Quick reference for perfect squares and their square roots

n = 1

\sqrt{1} = 1

n = 2

\sqrt{4} = 2

n = 3

\sqrt{9} = 3

n = 4

\sqrt{16} = 4

n = 5

\sqrt{25} = 5

n = 6

\sqrt{36} = 6

n = 7

\sqrt{49} = 7

n = 8

\sqrt{64} = 8

n = 9

\sqrt{81} = 9

n = 10

100

\sqrt{100} = 10

n = 11

121

\sqrt{121} = 11

n = 12

144

\sqrt{144} = 12

n = 13

169

\sqrt{169} = 13

n = 14

196

\sqrt{196} = 14

n = 15

225

\sqrt{225} = 15

n = 16

256

\sqrt{256} = 16

n = 17

289

\sqrt{289} = 17

n = 18

324

\sqrt{324} = 18

n = 19

361

\sqrt{361} = 19

n = 20

400

\sqrt{400} = 20

Why Use Our Square Root Calculator?

Comprehensive Results

Shows both positive and negative roots with detailed explanations

Educational Focus

Learn manual calculation methods and mathematical concepts

Complex Numbers

Optional complex number results for negative inputs

Free & Fast

No registration required, instant accurate results

Learn More About Square Roots

Khan Academy

Video lessons on understanding square roots and radicals

OpenStax

Free textbook chapter on simplifying and using square roots

Math is Fun

Interactive guide to square roots with visual examples

Frequently Asked Questions

What is a square root?

A square root of x is a value y such that y² = x. Every positive number has two square roots: a positive (principal) root and a negative root. Example: √9 = 3 because 3² = 9, and −3 is also a square root because (−3)² = 9.

What is the principal square root?

The principal square root is the non-negative square root of a number, denoted by √. When we write √25 = 5, we refer to the principal root. The full solution set is ±5, but √ alone denotes only the positive value.

Can negative numbers have square roots?

In the real number system, negative numbers do not have real square roots because no real number squared is negative. In the complex numbers, √−1 = i, so √−9 = 3i. The calculator can show complex results.

What is a perfect square?

A perfect square is a number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Perfect squares have integer square roots, making calculations simpler.

How do I calculate square roots manually?

Three classical methods: 1) prime factorization — factor the number and pair identical factors; 2) long-division style digit-by-digit extraction; 3) Newton's method — iterate x_{n+1} = (x_n + a/x_n)/2. The calculator shows each step.

Radical Calculator

Simplify square roots, cube roots, and nth roots with step-by-step solutions using prime factorization.

Pythagorean Theorem Calculator

Solve right triangle problems using a² + b² = c² with step-by-step solutions and visual triangles.

Quadratic Formula Calculator

Solve quadratic equations ax²+bx+c=0 using the quadratic formula with step-by-step solutions and discriminant analysis.

Simplify Calculator

Simplify fractions, radicals, and mathematical expressions with step-by-step solutions.