Square Root Calculator

Roots, powers, radical simplification, and prime factorization. Results update instantly.

Square Root (n=2)
8.485281374
sqrt(72) = 8.485281374...

Roots

Cube Root (n=3)
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Nth Root (n = 2)
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Powers

Squared (n2)
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Cubed (n3)
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Nth Power (n = 2)
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Perfect Square?
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Simplified Radical Form

Simplified sqrt(n)
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Prime Factorization

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How to Use the Square Root Calculator

Enter any positive number in the Number field and an optional value for n (the root degree and power exponent). All results — roots, powers, radical simplification, and prime factorization — update instantly as you type.

Understanding Square Roots

The square root of a number x is the value y such that y multiplied by y equals x. Written as sqrt(x) or x^(1/2), the square root is one of the most fundamental operations in mathematics. Every positive number has exactly one positive real square root, called the principal square root.

Square roots appear constantly in geometry (the Pythagorean theorem: c = sqrt(a^2 + b^2)), in physics (root-mean-square velocity, standard deviation), in statistics (standard deviation = sqrt(variance)), and in everyday calculations like finding the side length of a square from its area.

Cube Roots and Nth Roots

The cube root of x is the value y such that y^3 = x. For example, the cube root of 27 is 3, because 3 x 3 x 3 = 27. Cube roots are essential in volume calculations — if a cube has volume V, each side has length equal to the cube root of V.

More generally, the Nth root of x is the value y such that y^N = x. This calculator lets you specify any positive integer N. Enter N = 4 to find fourth roots, N = 5 for fifth roots, and so on. Fractional exponents are equivalent: the Nth root of x equals x raised to the power (1/N).

Perfect Squares

A perfect square is an integer whose square root is also a whole integer. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Perfect squares arise naturally in geometry (square grid areas), combinatorics, and number theory. This calculator checks whether your input is a perfect square by computing the square root, rounding to the nearest integer, and checking whether squaring that integer gives back the original number.

Simplifying Radicals

A radical expression like sqrt(72) can often be simplified by pulling out perfect square factors. The process: find the prime factorization, group prime factors into pairs, and bring each pair outside the radical as its square root. For sqrt(72): the prime factorization is 2^3 x 3^2. Group pairs: (2^2) x (3^2) x 2. Pull out 2 and 3: 2 x 3 = 6, leaving 2 inside. Result: 6 sqrt(2).

A simplified radical is in its simplest form when no perfect square factor remains inside the radical. For example, sqrt(50) = 5 sqrt(2) because 50 = 25 x 2 = 5^2 x 2. Simplified radical form is exact — no decimal approximation — making it useful for algebra, trigonometry, and calculus.

Prime Factorization

Prime factorization expresses a positive integer as a product of prime numbers. Every integer greater than 1 has a unique prime factorization (the Fundamental Theorem of Arithmetic). For example, 360 = 2^3 x 3^2 x 5. Prime factorizations are the foundation for simplifying radicals, finding least common multiples, reducing fractions, and many cryptographic algorithms.

This calculator uses trial division to find all prime factors of the input number (up to 10 million). The result is displayed both as a product with multiplication signs and in exponential notation where factors repeat.

Powers: Squares, Cubes, and Nth Powers

While roots ask "what number multiplied by itself N times gives x?", powers ask the reverse: "what do we get when we multiply x by itself N times?" The square of x (x^2) and the cube of x (x^3) are the most commonly used powers. This calculator shows x^2, x^3, and x^n for any n you specify, letting you quickly explore the relationship between a number and its roots and powers.

FAQ

A perfect square is an integer whose square root is also an integer. For example, 9 is a perfect square because sqrt(9) = 3 exactly. The sequence of perfect squares is 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on. Numbers like 2, 3, 5, 7, and 10 are not perfect squares because their square roots are irrational.
Find the largest perfect square factor of the number, then split the radical. For sqrt(72): 72 = 36 x 2, and 36 is a perfect square (6^2). So sqrt(72) = sqrt(36 x 2) = sqrt(36) x sqrt(2) = 6 sqrt(2). Using prime factorization: 72 = 2^3 x 3^2. Group prime pairs and pull them out: 2 x 3 = 6 with one 2 remaining inside, giving 6 sqrt(2).
The Nth root of a number x is the value y such that y^N = x. The square root is the 2nd root and the cube root is the 3rd root. For example, the 4th root of 81 is 3 because 3^4 = 81. In this calculator, enter the root degree in the n field to see any Nth root. Nth roots are computed as x^(1/N).
Prime factorization breaks a number into its prime number building blocks, and is used for: simplifying radicals (pull out perfect square prime pairs), reducing fractions (cancel common prime factors from numerator and denominator), finding the greatest common divisor (GCD) and least common multiple (LCM), and in cryptography (RSA encryption relies on the difficulty of factoring large numbers).
No real number, when multiplied by itself, gives a negative result: positive x positive = positive, and negative x negative = also positive. Square roots of negative numbers require imaginary numbers, where i = sqrt(-1). For example, sqrt(-4) = 2i. This calculator works with non-negative real numbers. If you need complex number arithmetic, use the scientific calculator.

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