Square Root Calculator

Calculate square roots with step-by-step solutions, simplified radical form, complex numbers for negatives, and approximation methods.

Formula:sqrt(x) = y where y^2 = x

Result

Square Root

12

Input144
Perfect SquareYes

Enter a Number

Enter any number (including negatives for complex results)

Number of decimal places (2-15)

Every positive number has two square roots

sqrt(144) = 12

Step-by-Step Solution

1

Given number: 144

2

144 is a perfect square.

3

sqrt(144) = 12

4

Verification: 12^2 = 144 = 144

(12)^2 = 144.0000000000 (approximately 144)

Result

Square Root

12

Input144
Perfect SquareYes

?How Do You Calculate a Square Root?

The square root of x (written sqrt(x) or x^(1/2)) is the number that, when multiplied by itself, gives x. For example, sqrt(25) = 5 because 5 x 5 = 25. Every positive number has two square roots: positive and negative (sqrt(25) = +5 and -5). Negative numbers have imaginary square roots: sqrt(-1) = i. Non-perfect squares can be simplified: sqrt(72) = 6sqrt(2).

What is a Square Root?

A square root of a number x is a value y such that y^2 = x. Written as sqrt(x), radical(x), or x^(1/2). The principal (positive) square root is typically implied when we write sqrt(x). Square roots are fundamental in solving quadratic equations, the Pythagorean theorem, standard deviation calculations, and many other mathematical applications.

Key Facts About Square Roots

  • sqrt(x) is the number y where y^2 = x
  • sqrt(25) = 5 because 5 x 5 = 25
  • Every positive number has two roots: +sqrt(x) and -sqrt(x)
  • Perfect squares have integer roots: sqrt(144) = 12
  • Radicals can be simplified: sqrt(72) = sqrt(36 x 2) = 6sqrt(2)
  • Square root of negative = imaginary: sqrt(-9) = 3i
  • sqrt(0) = 0 and sqrt(1) = 1
  • sqrt(a x b) = sqrt(a) x sqrt(b)

Frequently Asked Questions

A square root of a number x is a value y such that y x y = x. For example, the square root of 25 is 5 because 5 x 5 = 25. Every positive number has two square roots: a positive one (principal) and a negative one.
Not in the real number system. The square root of a negative number is an imaginary number. For example, sqrt(-1) = i, and sqrt(-9) = 3i. These complex numbers are used extensively in engineering and physics.
To simplify sqrt(n), find the largest perfect square factor of n, then sqrt(ab) = sqrt(a) x sqrt(b). For example: sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2) = 6sqrt(2).
A perfect square is a number whose square root is an integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Newton-Raphson is an iterative algorithm to find square roots: start with a guess g, then repeatedly compute g = (g + n/g) / 2. Each iteration gets closer to sqrt(n). It converges very quickly.

Last updated: 2025-01-15