The square root of a number is the value that, multiplied by itself, gives that number: √49 = 7 because 7 × 7 = 49. Most numbers are not perfect squares, so their roots are irrational decimals that never end. Mathematicians handle those by writing them in simplest radical form, such as √72 = 6√2, which is exact and often easier to work with than a long decimal. This calculator gives you both.
How to use the square root calculator
- Type the number under the radical (the radicand). Whole numbers, decimals like 2.25, fractions like 9/16 and negative numbers are all accepted.
- The tape shows the decimal value, the simplest radical form, whether the number is a perfect square and the two whole numbers the root lies between.
- The steps show the prime factorization and how pairs of factors come out of the radical.
Square root formulas and rules
| Rule | Formula | Example |
|---|---|---|
| Product | √(ab) = √a · √b | √72 = √36 · √2 = 6√2 |
| Quotient | √(a/b) = √a / √b | √(9/16) = 3/4 |
| Rationalizing | 1/√a = √a / a | 1/√2 = √2/2 |
| Negative radicand | √(−a) = i√a | √−16 = 4i |
| As an exponent | √x = x^(1/2) | 25^(1/2) = 5 |
The product rule is the engine of simplification: split the radicand into a perfect square times what remains.
Worked example: √72
Factor: 72 = 2³ × 3² = (2² × 3²) × 2.
Pull out pairs: each pair of equal factors leaves the radical as a single factor, so 2 and 3 come out: 2 × 3 = 6, and one 2 stays inside.
Result: √72 = 6√2 ≈ 6 × 1.4142135624 = 8.4852813742.
Sense check: 8² = 64 and 9² = 81, and 72 lies between them, so the root is between 8 and 9. ✓
Similarly √50 = 5√2 ≈ 7.0710678119 and √0.5 = √2/2 ≈ 0.7071067812.
Estimating square roots by hand
Bracket and interpolate
Find the perfect squares on either side. For √55: 49 < 55 < 64, so the root is between 7 and 8. Since 55 is 6 of the 15 steps from 49 to 64, a first guess is 7 + 6/15 ≈ 7.4. The true value is 7.416.
The Babylonian method
This ancient method, a special case of Newton’s method, doubles the number of correct digits with each round. Start with any guess g, then replace it with the average of g and x/g:
For √2, starting from 1.5: (1.5 + 2/1.5)/2 = 1.41667, then (1.41667 + 2/1.41667)/2 = 1.4142157, already correct to five decimal places. A Babylonian clay tablet known as YBC 7289, about 3,800 years old, records √2 to the equivalent of six decimal places.
Where square roots appear
- Geometry. The diagonal of a square with side s is s√2, and the Pythagorean theorem gives c = √(a² + b²).
- Distance. The straight-line distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²).
- Statistics. Standard deviation is the square root of the variance.
- Physics. Falling-object time is t = √(2h/g), and the period of a pendulum grows with √L.
For cube, fourth and higher roots, see the cube root calculator and the nth root calculator. To simplify a radical with a coefficient in front, such as 3√72, use the simplify radicals calculator.
Frequently asked questions
What is the simplified form of a square root?
A square root is in simplest form when no perfect square other than 1 divides the number under the radical, and no fraction or radical remains in a denominator. For example √72 simplifies to 6√2 because 72 = 36 × 2 and √36 = 6.
Does a number have two square roots?
Every positive number has two: one positive and one negative, since both 8.485… and −8.485… square to 72. The symbol √ means the positive one, called the principal square root, so √72 ≈ 8.485 while x² = 72 has solutions x = ±√72.
Can you take the square root of a negative number?
Not with real numbers, because every real number squared is zero or positive. Using the imaginary unit i = √−1, √−16 = 4i and √−72 = 6√2·i.
How do I take the square root of a fraction?
Take the root of the top and bottom separately, then remove the radical from the denominator. √(9/16) = 3/4 exactly, while √(1/2) = 1/√2 = √2/2 ≈ 0.7071.
How accurate is the decimal?
Decimals are shown to ten places. Perfect squares, including fractions such as 9/16 and decimals such as 2.25, are detected exactly and shown without rounding.