Square Root Calculator
Calculate square roots, cube roots, and nth roots — with radical simplification
Common Perfect Squares
Root Properties
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What Is a Square Root?
A square root is the inverse operation of squaring a number. If you square 5 you get 25, so the square root of 25 is 5. More formally, √x is the number that, when multiplied by itself, gives x. Square roots show up everywhere in real life: calculating the side of a square given its area, applying the Pythagorean theorem to find diagonal distances, evaluating physics formulas like velocity from kinetic energy, and computing statistical measures such as standard deviation.
Not all square roots are tidy whole numbers. When the number under the radical is a perfect square (1, 4, 9, 16, 25 …) the result is a rational integer. Every other positive number produces an irrational square root — a non-repeating, non-terminating decimal like √2 ≈ 1.41421356… Radical simplification rewrites these results in their simplest exact form, for example √72 = 6√2, which is more useful in algebra than the raw decimal.
How to Use This Calculator
- 1Enter the number you want to find the root of in the input field.
- 2Optionally enter the root index — leave it blank or set it to 2 for a square root, 3 for a cube root, or any positive integer for an nth root.
- 3Click Calculate (or press Enter) to compute the result.
- 4Read the exact simplified radical form and the full decimal approximation shown below the button.
Formulas & Reference
Square root: √x = x^(1/2)
Cube root: ∛x = x^(1/3)
Nth root: ⁿ√x = x^(1/n)
Simplifying radicals:
√72 = √(36×2) = 6√2
√50 = √(25×2) = 5√2
Pythagorean theorem:
c = √(a² + b²)
Example: a=3, b=4 → c = √(9+16) = √25 = 5
Distance formula:
d = √((x₂−x₁)² + (y₂−y₁)²)Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …) have exact integer square roots. Every other positive number has an irrational square root — a decimal that never repeats and never terminates. Negative numbers have no real square root; their roots are imaginary numbers written with the unit i.
Worked Examples
Perfect square vs. irrational root
√144 = 12 exactly, because 12 × 12 = 144 — this is a perfect square. By contrast, √2 ≈ 1.41421 is irrational: no fraction can express it exactly, and its decimal expansion goes on forever without repeating.
Pythagorean theorem: finding a hypotenuse
A right triangle has legs of length 5 and 12. The hypotenuse c = √(5² + 12²) = √(25 + 144) = √169 = 13. Because 169 is a perfect square, the answer is a clean whole number — a classic Pythagorean triple.
Simplifying a radical: √180
Factor 180 into the largest perfect-square factor: 180 = 36 × 5. Then √180 = √36 × √5 = 6√5. In decimal form, 6√5 ≈ 6 × 2.2361 ≈ 13.416. The simplified radical form 6√5 is exact and much more useful in algebraic work.