Square Root Calculator
Find the square root of any number, fraction or negative number. Get the simplified radical form like 6√2, up to 100 decimal places, and the steps.
How Square Roots Work
The square root of x is a number that gives x when you multiply it by itself. √49 = 7 because 7 × 7 = 49. Every positive number has two square roots, one positive and one negative (7 and −7), but the symbol √ means the positive one, called the principal square root.
To simplify √x, split x into prime factors and take each pair of equal primes out of the root. For 72 = 2 × 2 × 2 × 3 × 3, the pairs 2 × 2 and 3 × 3 come out as 2 and 3, and one 2 stays inside: √72 = 2 × 3 × √2 = 6√2. A root with no pairs left inside is in simplest form.
For a fraction, take the root of the top and bottom: √(9/16) = 3/4. If the bottom is not a perfect square, multiply top and bottom by its root so the answer has no root below the line: √(8/3) = 2√6/3. A negative number has no real square root, because any real number squared is 0 or more. Its roots are imaginary: √−12 = 2i√3, where i² = −1.
Disclaimer: This calculator performs standard mathematical operations for general and educational use only. We are not a substitute for professional review — for calculations where accuracy is critical, such as financial filings, engineering work, or academic submissions, please verify the result independently.
Frequently Asked Questions
About this calculator
This square root calculator gives you the square root of any number in two ways: the exact simplified form, such as √72 = 6√2, and a decimal value with up to 100 decimal places. It also works with fractions (√(8/3) = 2√6/3), decimals (√0.25 = 1/2) and negative numbers (√−12 = 2i√3). For each answer it shows the steps, both square roots, whether the number is a perfect square, the perfect squares on each side, and a check by squaring the result.
What is a square root?
The square root of a number x is the number that gives x when you multiply it by itself. √49 = 7 because 7 × 7 = 49. Every positive number has two square roots, a positive one and a negative one (7 and −7). The symbol √ always means the positive one, called the principal square root. A perfect square is a whole number whose square root is also a whole number, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and 144.
How the calculator works
Simplified radical form. The calculator splits the number into prime factors and takes each pair of equal primes out of the root. 72 = 2³ × 3², so the pairs 2 × 2 and 3 × 3 come out as 2 and 3, and one 2 stays inside: √72 = 2 × 3 × √2 = 6√2. A root is in simplest form when no square number (other than 1) divides the number under the root.
Fractions and decimals. A decimal is first written as a fraction (2.25 = 9/4). Then √(a/b) = √a / √b. If the bottom is not a perfect square, the calculator multiplies the top and bottom by its root, so no root is left in the denominator. This is called rationalizing: √(8/3) = √8 / √3 = 2√2 / √3 = 2√6/3.
Negative numbers. No real number squared gives a negative number, so a negative number has no real square root. Its square roots are imaginary numbers. The calculator takes out √−1 = i: √−12 = i × √12 = 2i√3, because (2i√3)² = 4 × i² × 3 = −12.
High precision. The decimal value is not computed with ordinary floating-point numbers, which only keep about 16 digits. Instead the calculator finds the whole-number square root of x × 10²ᵈ with exact integer arithmetic, so every one of the up to 100 decimal places is correct, and the last place is rounded. For example, √2 = 1.4142135624 to 10 places.
Worked examples
- √72: 72 = 2³ × 3², so √72 = 6√2 ≈ 8.485281374. Check: (6√2)² = 36 × 2 = 72. It lies between 8 and 9, because 64 < 72 < 81.
- √2025: 2025 = 3⁴ × 5², so √2025 = 3² × 5 = 45. It is a perfect square.
- √(8/3): = 2√6/3 ≈ 1.632993. Check: (2√6/3)² = 4 × 6 / 9 = 8/3.
- √0.25: 0.25 = 1/4, so √0.25 = 1/2 = 0.5.
- √−12: = 2i√3 ≈ 3.464102i. The two square roots are ±2i√3.
The Babylonian method
Calculators and computers find square roots with a method that is about 4,000 years old. Start with a guess g, then replace it with the average of g and x/g, and repeat. For √72, start with g = 8: the next guesses are (8 + 72/8) / 2 = 8.5, then 8.485294, then 8.4852813742. After 4 steps the answer is correct to all 16 digits a computer normally keeps. The calculator shows this table for every number that is not a perfect square, so you can follow how fast the guesses close in.
Square roots in everyday life
- Area to side length: a square room with an area of 50 m² has sides of √50 = 5√2 ≈ 7.07 m.
- Diagonals: by the Pythagorean theorem, the diagonal of a door 0.9 m wide and 2.0 m high is √(0.9² + 2.0²) = √4.81 ≈ 2.19 m. This tells you whether a long table can pass through at an angle.
- Falling objects: an object dropped from 20 m takes t = √(2h/g) = √(2 × 20 / 9.81) ≈ 2.02 seconds to hit the ground.
- Statistics: the standard deviation is the square root of the variance. A variance of 16 gives a standard deviation of √16 = 4.
This tool is for square roots only. For cube roots and other roots, such as ∛27 or the 4th root of 81, use the Root Calculator in the Mathematics category.
- Simplified radical form — Gives the exact answer in simplest form, such as √72 = 6√2 or √(8/3) = 2√6/3, with the prime-factor steps.
- Up to 100 decimal places — Calculates the decimal value with exact integer arithmetic, so all 10, 20, 50 or 100 places are correct.
- Fractions, decimals and negatives — Works with inputs like 8/3, 0.25 or −12, and gives imaginary roots (such as 2i√3) for negative numbers.
- Perfect square check — Tells you if the number is a perfect square and shows the perfect squares on each side, such as 8 < √72 < 9.
- Babylonian method table — Shows how the classic guess-and-average method closes in on the square root, step by step.