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Square Root Calculator

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

Can be positive, negative, or zero

Try These Examples
Click on any example to automatically calculate
Example
25\sqrt{25}

Perfect square example

Example
12\sqrt{12}

Non-perfect square

Example
0\sqrt{0}

Square root of zero

Example
−4\sqrt{-4}

Negative number (complex result)

Example
144\sqrt{144}

Larger perfect square

Example
2\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 y2=xy^2 = x, then y is a square root of x.

Key Concepts:

  • Principal Root: The non-negative square root (x\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: 9=3\sqrt{9} = 3 because 32=93^2 = 9
But 9 also has a negative square root: -3, because (−3)2=9(-3)^2 = 9

How to Calculate Square Roots Manually?

Method 1: Prime Factorization

  1. Factor the number into prime factors
  2. Group identical factors in pairs
  3. Take one number from each pair outside the radical
  4. Multiply the numbers outside

Method 2: Long Division Method

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

Method 3: Estimation & Refinement

  1. Find perfect squares above and below
  2. Make an initial estimate
  3. Use Newton's method: xn+1=12(xn+axn)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 = √(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
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

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