Square Root Calculator

Compute principal square roots, simplify radicals into a√b form, and practice estimation.

Number (X)72
0500
Principal Square Root (√72)
8.485281

The value that, when multiplied by itself, yields exactly 72.

Radical Simplification
√72
=
6√2

Simplified by extracting the perfect square factor 36 (since √36 = 6).

Perfect Squares Number Line
366²497²648²819²10010²√72
How is this calculated?

To calculate √72:

  1. We find the principal square root value: √72 = 8.485281.
  2. We find the largest perfect square factor of 72, which is 36 (since 6² = 36). We write √72 = √(36 × 2) = 6√2.
Last Verified: 2026-07-11Sources: Wolfram MathWorld, Algebra Textbooks

About the Square Root Calculator

A square root of a number is a value that, when multiplied by itself, yields the original number. In algebra and geometry, the positive or non-negative square root of a real number is known as the principal square root, represented by the radical symbol √. While every positive number has two real square roots (one positive and one negative), the principal root is the standard mathematical output. Real numbers can be divided into perfect squares, which yield integer roots (e.g., √25 = 5), and non-perfect squares, which yield irrational numbers (e.g., √2 ≈ 1.414213) that have an infinite, non-repeating decimal expansion. Square roots are fundamental in computing geometric distances (such as the Pythagorean theorem), estimating statistical distributions, and solving quadratic equations. In computer science and calculators, square roots are computed using efficient numerical approximations like Newton-Raphson iterations or the ancient Babylonian method.

Mathematical Formula & Logic

Square roots and radical simplifications are modeled using algebraic properties of exponents and radicals: 1. Mathematical Identity: If y = √x, then y² = x (where x, y ≥ 0 for real numbers) 2. Exponential Equivalence: √x = x^(0.5) = x^(1/2) 3. Radical Product Rule (used for simplification): √(a·b) = √a · √b 4. Simplest Radical Form (a√b): If x = a² · b (where a² is the largest perfect square factor of x), then: √x = √(a²·b) = a√b 5. Babylonian Method (Heron's Iterative Formula): x_(n+1) = 0.5 × (x_n + S / x_n) Where S is the input number, and x_n is the current approximation (guess).

Step-by-Step Example

Simplify the radical √72 into its simplest radical form and calculate its decimal approximation: 1. Find all perfect square factors of 72: - Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 - Perfect square factors: 1, 4, 9, 16 (no), 25 (no), 36 - The largest perfect square factor is 36. 2. Express 72 as a product of its largest perfect square factor and a remainder: 72 = 36 × 2 3. Apply the product rule for radicals: √72 = √(36 × 2) = √36 × √2 = 6√2 4. Compute the decimal approximation using 6√2: - √2 ≈ 1.41421356 - 6 × 1.41421356 ≈ 8.485281 5. The simplest radical form of √72 is 6√2, which approximates to 8.485281. Estimating without a calculator Trap the number between the two perfect squares either side of it. 72 sits between 64 and 81, so √72 lies between 8 and 9, and closer to 8 because 72 is nearer 64. That alone is usually enough to catch a mistyped answer. For a real digit-by-digit answer, use the Babylonian method - the iteration in the formula section, known to Heron of Alexandria and to Babylonian scribes long before him. Take any first guess and repeatedly average it with the number divided by that guess. Starting at 8 for √72: x₁ = ½(8 + 72/8) = 8.5 x₂ = ½(8.5 + 72/8.5) = 8.4852941176 x₃ = ½(8.4852941176 + 72/8.4852941176) = 8.4852813742 The true value is 8.4852813742. Three steps from a deliberately rough guess produce ten correct digits, because the method roughly doubles the number of accurate digits each iteration. This is essentially what the hardware inside your calculator does. Three errors worth avoiding The root does not distribute over addition. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. It does distribute over multiplication and division, which is exactly why the simplification above works. √(x²) is |x|, not x. Squaring destroys the sign and the radical cannot recover it, so √((−5)²) = √25 = 5, not −5. This is the step that silently loses solutions when you solve equations. A negative number has no real square root. No real value multiplied by itself gives −9, since two negatives make a positive. Such roots exist only in the complex numbers, where √(−9) = 3i.

Reference Data & Values

inputradical formsimplifieddecimallies between
422 (perfect square)2.000000exact
12√122√33.4641023 and 4
18√183√24.2426414 and 5
50√505√27.0710687 and 8
72√726√28.4852818 and 9
98√987√29.8994959 and 10
147√1477√312.12435612 and 13
200√20010√214.14213614 and 15
250√2505√1015.81138815 and 16

Frequently Asked Questions

By strict mathematical definition, the radical symbol √ represents the principal square root function, which is defined to return only the non-negative root. This convention ensures that the square root is a well-defined function (where each input has exactly one output). If you need both roots (e.g., when solving x² = 9), you must write ±√9 to denote both +3 and -3.
The Babylonian method is an ancient iterative algorithm. To find the square root of S, you make an initial guess (e.g., x_0). You then calculate a new guess by taking the average of your current guess and S divided by your current guess: x_1 = 0.5 * (x_0 + S / x_0). This process converges quadratically, meaning the number of correct decimal places approximately doubles with each step.
In the set of real numbers, you cannot calculate the square root of a negative number because any real number multiplied by itself is always non-negative. However, in the complex number system, the square root of a negative number is defined using the imaginary unit i (where i = √-1). For example, √-16 is expressed as 4i.
A perfect square is an integer that can be expressed as the square of another integer. For example, 9 is a perfect square because 3 × 3 = 9. The square root of any perfect square is always an integer.
To simplify the square root of a fraction, apply the quotient property of radicals: √(a/b) = √a / √b. Simplify the numerator and denominator separately, and if necessary, rationalize the denominator to remove any radical symbols.
No, the square root of 2 is an irrational number, which was first proven by the Pythagoreans. Its decimal representation (1.41421356...) continues infinitely without repeating patterns, and it cannot be expressed as a simple fraction.
The principal square root is the non-negative square root of a non-negative real number. While x² = 9 has two solutions (+3 and -3), the principal square root of 9 is strictly the positive value +3.
Because the square root distributes over multiplication and division, not over addition or subtraction. √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. The two differ because squaring a sum produces a cross term - (a + b)² = a² + 2ab + b² - and the root has no way to undo that middle term. This is the most common error in radical algebra, and the reason simplification works by factoring into products rather than splitting sums.
Squaring discards the sign, and the radical returns the principal (non-negative) root, so it cannot put the sign back. √((−5)²) = √25 = 5, not −5. Writing √(x²) = x is only safe when x is already known to be non-negative; otherwise the absolute value is required, and forgetting it is how solutions quietly disappear when solving equations.
Trap it between the perfect squares either side. For √50, note that 49 = 7² and 64 = 8², so the answer lies between 7 and 8, and much nearer 7 because 50 is barely above 49. The true value is 7.071068. For more precision, average your guess with the number divided by the guess and repeat - the Babylonian method roughly doubles the number of correct digits each time, reaching ten digits for √72 in three steps from a first guess of 8.
Because 0 is the only number that, when multiplied by itself, yields 0 (0 × 0 = 0). Thus, √0 = 0.