Square Root Calculator
Calculate the square root of positive and negative numbers (√x). Provides radical simplification, perfect square check, and imaginary roots.
Did this calculation save you time?
Student ProjectHi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.
I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!
What Is a Square Root Calculator?
A square root of a number x is a number y such that y² = x. Every positive real number has two square roots (one positive and one negative); the non-negative one is called the principal square root, denoted √x.
How to Use This Calculator
- Enter any positive or negative real number.
- View the principal real square root or complex root (e.g. ±4i).
- Review the simplified radical notation and step-by-step breakdown.
Square Root Definition
y = \sqrt{x} \iff y^2 = x \quad (y \ge 0), \quad \sqrt{-x} = i\sqrt{x}The radical symbol denotes the principal non-negative square root. For negative numbers, i represents the imaginary unit √(-1).
Worked Example
Scenario: Find square root of 72
Given: x = 72
Decimal root: √72 ≈ 8.485281
Factor into primes: 72 = 36 × 2
Simplify radical: √(36 × 2) = √36 × √2 = 6√2
Tips & Key Notes
- For negative numbers, the square root involves the imaginary unit: √(-16) = 4i.
- To simplify radicals by hand, factor out the largest perfect square (4, 9, 16, 25, 36, etc.).
- In exponential notation, √x is written as x^(1/2).
Frequently Asked Questions
Can you take the square root of a negative number?
Not within real numbers, but in complex numbers: √(-a) = i√a, where i = √(-1). For example, √(-25) = 5i.
Why does √25 only equal 5 and not -5?
By mathematical convention, the radical sign √ denotes the principal (non-negative) root. The equation x² = 25 has two solutions (±5), but √25 = 5.
What is an irrational square root?
If a whole number is not a perfect square, its square root is an irrational number with non-repeating, non-terminating decimals (like √2 ≈ 1.4142).
How do you estimate square roots without a calculator?
Find the two closest perfect squares. For √50: 7² = 49 and 8² = 64, so √50 is slightly greater than 7 (≈ 7.07).
Related Calculators
Explore similar toolsSquare Calculator (x²)
Calculate the square of any positive or negative number (x² = x × x) with scientific notation and step-by-step arithmetic.
Cube Root Calculator
Calculate the principal real cube root of positive and negative numbers. Identifies perfect cubes, nearest integer roots, and complex conjugate roots.
Nth Root Calculator
Calculate the nth root of any number (ⁿ√x). Supports even and odd roots, real roots, radical expressions, and step-by-step solutions.