Calculate square roots with step-by-step solutions, simplified radical form, complex numbers for negatives, and approximation methods.
Square Root
12
Enter any number (including negatives for complex results)
Number of decimal places (2-15)
Every positive number has two square roots
sqrt(144) = 12
Given number: 144
144 is a perfect square.
sqrt(144) = 12
Verification: 12^2 = 144 = 144
(12)^2 = 144.0000000000 (approximately 144)
Square Root
12
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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).
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.
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
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Square Root
12