Square Root Calculator - Any nth Root + Factors

The square root of a number is the value that, when multiplied by itself, produces the original number.

The square root of 25 is 5 because 5 x 5 = 25. While perfect squares like 25, 36, and 100 have clean integer roots, most numbers produce irrational roots with infinite decimal places, which is where this square root calculator becomes essential.

Enter any positive number and the calculator returns the square root as a decimal, as a simplified radical (when applicable), and identifies whether the number is a perfect square. It also supports nth roots (cube root, fourth root, and beyond), allowing you to solve a wide range of radical expressions from a single tool.

Whether you are solving geometry problems, simplifying radical expressions in algebra, checking a manual calculation, or computing distances using the Pythagorean theorem, this calculator provides precise results instantly. Use it alongside the Exponent Calculator for related power and root calculations.

What Is a Square Root

The square root of a number x is the value y such that y x y = x. Symbolically, y = sqrt(x). Every positive number has two square roots: one positive and one negative. The square root of 49 is both +7 and -7, since 7 x 7 = 49 and (-7) x (-7) = 49. By convention, the symbol sqrt refers to the positive (principal) root.

Perfect squares are integers whose square roots are also integers:

sqrt(1) = 1
sqrt(4) = 2
sqrt(9) = 3
sqrt(16) = 4
sqrt(25) = 5
sqrt(36) = 6
sqrt(49) = 7
sqrt(64) = 8
sqrt(81) = 9
sqrt(100) = 10

Numbers that are not perfect squares have irrational square roots, meaning their decimal representations go on forever without repeating. The square root of 2 is approximately 1.41421356…, and the square root of 3 is approximately 1.73205080…. These values cannot be expressed as exact fractions.

Simplifying Square Roots

When a number is not a perfect square, its square root can sometimes be simplified by factoring out perfect square components.

To simplify sqrt(n):

Find the largest perfect square factor of n.
Rewrite n as the product of that factor and the remaining factor.
Take the square root of the perfect square factor.

Example: Simplify sqrt(72).

72 = 36 x 2, and 36 is a perfect square.
sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2).
sqrt(72) = 6 x sqrt(2), or 6sqrt(2).

Example: Simplify sqrt(200).

200 = 100 x 2.
sqrt(200) = sqrt(100) x sqrt(2) = 10sqrt(2).

This calculator automatically simplifies roots when possible and presents both the simplified radical form and the decimal approximation.

Properties of Square Roots

Several algebraic properties make square roots easier to work with:

Product rule: sqrt(a x b) = sqrt(a) x sqrt(b). This is the property used in simplification above.

Quotient rule: sqrt(a / b) = sqrt(a) / sqrt(b). For example, sqrt(25/9) = 5/3.

Power relationship: sqrt(x) = x^(1/2). This connects square roots to exponents and is fundamental in algebra and calculus.

Squaring reverses the root: (sqrt(x))^2 = x. This is often used to eliminate square roots from equations.

No real root for negatives: The square root of a negative number is not a real number. sqrt(-1) is defined as the imaginary unit i in complex number theory, but in standard real-number arithmetic, negative inputs have no square root. This calculator works with positive numbers and zero.

Nth Roots

This calculator extends beyond square roots to compute any nth root. The nth root of a number x is the value y such that y^n = x.

Cube root (n=3): The cube root of 27 is 3 because 3^3 = 27. Unlike square roots, cube roots exist for negative numbers: the cube root of -8 is -2 because (-2)^3 = -8.

Fourth root (n=4): The fourth root of 81 is 3 because 3^4 = 81.

Fifth root (n=5): The fifth root of 32 is 2 because 2^5 = 32.

The general formula is: nth root of x = x^(1/n).

This is useful in many applied contexts. The geometric mean of n values involves taking the nth root. Annual growth rates derived from multi-year totals require nth roots. Decay calculations in science use nth roots to find per-period rates.

Methods for Estimating Square Roots

Before calculators, several methods existed for estimating square roots by hand.

The bracket method. Find two consecutive integers whose squares bracket the target number. sqrt(50) falls between 7 (49) and 8 (64), closer to 7. A rough estimate is 7.1. Test: 7.1^2 = 50.41, which is close.

The Babylonian method (Newton’s method). Start with an initial guess, then repeatedly improve it using the formula:

new guess = (guess + n/guess) / 2

For sqrt(50), starting with guess = 7:

Iteration 1: (7 + 50/7) / 2 = (7 + 7.143) / 2 = 7.071
Iteration 2: (7.071 + 50/7.071) / 2 = 7.0711

After just two iterations, the estimate is accurate to four decimal places. This method converges very quickly and is the basis of how many computer algorithms calculate square roots.

Prime factorization method. Factor the number into primes, then pair identical factors. Each pair contributes one factor to the root.

sqrt(1764) = sqrt(2 x 2 x 3 x 3 x 7 x 7) = 2 x 3 x 7 = 42

This method works perfectly for perfect squares and helps with simplification for non-perfect squares.

Square Roots in Geometry

Square roots appear throughout geometry, often in fundamental formulas.

Pythagorean theorem. In a right triangle with legs a and b and hypotenuse c: c = sqrt(a^2 + b^2). A triangle with legs of 3 and 4 has a hypotenuse of sqrt(9 + 16) = sqrt(25) = 5.

Distance formula. The distance between two points (x1, y1) and (x2, y2) is: d = sqrt((x2-x1)^2 + (y2-y1)^2). This is a direct application of the Pythagorean theorem. Use the Triangle Calculator for complete triangle solutions.

Area from side length. If you know the area of a square, the side length is the square root of the area. A square with an area of 144 square feet has sides of 12 feet. Use the Square Footage Calculator for practical area measurements.

Circle calculations. The radius of a circle given its area: r = sqrt(A / pi). A circle with an area of 100 square centimeters has a radius of approximately 5.64 cm.

Square Roots in Science and Engineering

Physics. Velocity from kinetic energy uses a square root: v = sqrt(2 x KE / m). The period of a pendulum involves sqrt(L/g). Wave speed, gravitational calculations, and many other physical relationships contain square roots.

Electronics. RMS (root mean square) voltage and current are calculated using square roots. Power calculations in AC circuits involve RMS values derived from peak values using sqrt(2).

Statistics. Standard deviation is the square root of variance. See the Standard Deviation Calculator for full statistical analysis.

Finance. Annualized volatility is calculated by multiplying daily volatility by the square root of the number of trading days (approximately sqrt(252)).

Frequently Asked Questions

What is the square root of 2?

The square root of 2 is approximately 1.41421356. It is an irrational number, meaning its decimal expansion is infinite and non-repeating. It is also known as Pythagoras’ constant and appears frequently in geometry, particularly in the diagonal of a unit square.

How do I simplify a square root?

Factor the number under the radical to find perfect square factors. For example, sqrt(48) = sqrt(16 x 3) = 4sqrt(3). Find the largest perfect square that divides evenly into your number, take its root, and leave the remaining factor under the radical.

Can you take the square root of a negative number?

Not in the real number system. The square root of a negative number is an imaginary number. sqrt(-1) is defined as i (the imaginary unit), so sqrt(-9) = 3i. This calculator works with real numbers only.

What is a perfect square?

A perfect square is an integer that is the square of another integer. Examples include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Their square roots are whole numbers.

What is the difference between a square root and a cube root?

A square root of x is the number that multiplied by itself gives x. A cube root of x is the number that multiplied by itself three times gives x. sqrt(27) is approximately 5.196, while the cube root of 27 is exactly 3 (since 3x3x3 = 27).

How is the square root symbol written?

The square root symbol is the radical sign, which looks like a checkmark with a horizontal bar extending over the number. In text, it is often written as sqrt(). For nth roots, a small number is placed in the notch of the radical: the cube root of 8 is written with a small 3 in the radical.

What is the square root of 0?

The square root of 0 is 0, since 0 x 0 = 0. Zero is the only number that is its own square root and is considered a perfect square.

Data accurate as of: March 2026