About the Square Root Calculator
This square root calculator finds √x for any number you enter, to up to 10 decimal places, and — for whole numbers — also writes it in simplest radical form, such as √72 = 6√2. It tells you whether the number is a perfect square and, if not, which two perfect squares it sits between, which is the quickest way to estimate a root by hand.
It is useful for algebra and geometry homework (the Pythagorean theorem, the quadratic formula, distances), for checking a mental estimate, and for anyone who needs the side length of a square from its area. The table shows the Babylonian (Newton) method converging on the answer step by step, so you can see how calculators actually compute square roots.
Every positive number has two square roots, a positive one and a negative one; the calculator shows the principal (positive) root and the ± pair. Negative numbers have no real square root, so their roots are shown as imaginary numbers using i = √−1.
With the default inputs, the square root (√x) is 8.4852813742. Change any value above to recalculate instantly.
How to use the square root calculator
- 1Enter the number you want the square root of.
- 2Read the decimal root and, for whole numbers, the simplest radical form.
- 3Check whether it is a perfect square and which perfect squares it lies between.
- 4Look at the iteration table to see how the answer is computed step by step.
Formula and method
The principal square root of a non-negative number x is the non-negative number y that, multiplied by itself, gives x. Calculators find it with the Babylonian (Newton) method: start with a guess g, replace it with the average of g and x ÷ g, and repeat. Each step roughly doubles the number of correct digits, so a handful of iterations gives full precision.
To simplify a radical, factor out the largest perfect square: √72 = √(36 × 2) = √36 × √2 = 6√2. A result is in simplest form when the number left under the root sign has no square factor other than 1. For a negative number, √(−x) = √x · i, where i is the imaginary unit.
- x
- The number whose root you want (the radicand)
- y
- Principal square root, y ≥ 0
- a√b
- Simplest radical form, b square-free
- i
- Imaginary unit, i² = −1
Worked examples
Square root of 72
72 = 36 × 2 and 36 is a perfect square, so √72 = 6√2 ≈ 8.4853. It lies between 8 (8² = 64) and 9 (9² = 81).
Perfect square 144
12 × 12 = 144, so 144 is a perfect square and its square root is exactly 12 (and −12 is the other root).
Square root of 2
2 has no square factor, so √2 is already in simplest form. Its decimal value 1.41421356… is irrational and never repeats.
Negative number −50
No real number squared gives −50. Since 50 = 25 × 2, √−50 = 5√2 · i ≈ 7.0711i, and the two roots are ±7.0711i.
Decimal 0.25
0.5 × 0.5 = 0.25, so √0.25 = 0.5. Square roots of numbers between 0 and 1 are larger than the number itself.
Frequently asked questions
How do you find a square root without a calculator?+
Find the two perfect squares either side of the number to estimate, then refine with the Babylonian method: divide the number by your guess and average the result with the guess. For √10: guess 3, 10 ÷ 3 = 3.333, average 3.1667 — already close to 3.1623.
What is a perfect square?+
A perfect square is a whole number that is another whole number multiplied by itself, such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and 144. Its square root is an exact integer.
How do you simplify a square root?+
Split the number into the largest perfect square times a leftover factor, then take the root of the perfect square outside: √48 = √(16 × 3) = 4√3. It is fully simplified when the number under the root has no square factor.
Can you take the square root of a negative number?+
Not with real numbers, because any real number squared is zero or positive. In complex numbers √−1 is defined as i, so √−9 = 3i and √−50 = 5√2 i.
Why does a number have two square roots?+
Both a positive and a negative number give the same square: 5² = 25 and (−5)² = 25. The √ symbol means the principal (non-negative) root, so √25 = 5, while the equation x² = 25 has solutions x = ±5.