Square Root Calculator
Calculate square roots, cube roots and nth roots with high precision
Enter a number to calculate its root
Square Root
A number y such that y² = x.
Cube Root
A number y such that y³ = x.
Nth Root
A number y such that y^n = x.
Newton's Method
An iterative method to find successively better approximations to the roots.
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What is a Square Root?
The square root of a number is the value that, when multiplied by itself, gives back the original number. The square root of 16 is 4, because 4 × 4 = 16. It is written with the radical symbol: √16 = 4. The number under the symbol is called the radicand.
Not every number has a tidy whole-number root. The square root of 2 (√2 ≈ 1.414213…) never terminates or repeats — it is an irrational number, discovered by the ancient Greeks. The radical symbol √ itself is thought to derive from a stylized lowercase “r” for the Latin radix (root), introduced in the 1500s.
The idea generalizes: a cube root (∛) reverses cubing (∛27 = 3), and an nth root reverses raising to the nth power. All roots are written with the same radical, with a small index number for roots beyond square.
How to Calculate Roots
For perfect squares, the root is simply memorized (√81 = 9, ∛64 = 4). For any other number, computers and calculators use an iterative method — most commonly the Newton–Raphson method — to approach the answer:
Newton’s method (for √N):
guess ₊ ₁ = (guess + N ÷ guess) ÷ 2
Start with a rough guess, plug it into the formula, and repeat. Each step roughly doubles the number of correct digits. After a handful of iterations, the guess stabilizes to 10+ decimals of accuracy — which is why your calculator answers instantly.
Examples:
√16 = 4 (because 4² = 16)
∛27 = 3 (because 3³ = 27)
√2 ≈ 1.41421356 (irrational — never ends)
Common Use Cases
Geometry — the Pythagorean theorem — The length of a right triangle’s hypotenuse is √(a² + b²). A ladder reaching 6 m up a wall, 8 m along the ground, is √(36 + 64) = √100 = 10 m long.
Statistics — standard deviation — Standard deviation, the most common measure of spread, is the square root of the variance. Converting variance back to the original units always requires a square root.
Engineering — RMS voltage — The “120V” wall outlet is really a Root-Mean-Square value: the effective voltage of an AC waveform, computed by squaring, averaging, then taking a square root.
Finance — volatility and risk — Variance of returns is easy to compute, but to interpret it in the original currency or percentage units you take its square root, yielding the standard deviation used to measure investment risk.
Frequently Asked Questions
What is a perfect square?
A perfect square is a whole number whose square root is also a whole number. Examples include 1 (1×1), 4 (2×2), 9 (3×3), 16 (4×4), 25, 36, 49, 64, 81, and 100. The first ten perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Numbers like 2 or 20 are not perfect squares because their square roots (√2 ≈ 1.414, √20 ≈ 4.472) are not integers.
What is the square root of a negative number?
In the real number system, the square root of a negative number is undefined, because no real number multiplied by itself gives a negative result. To handle them, mathematicians use imaginary numbers: the square root of −1 is defined as i, so √(−9) = 3i. This is the basis of the complex number system used in engineering and physics. Our real-number calculator rejects negative inputs for square roots.
How many decimal places do I need for a square root?
It depends on your use case. For quick estimates, 2–3 decimals are enough (√2 ≈ 1.41). For engineering and scientific work, 6–8 decimals are typical. Most calculators and our tool support up to 10–15 decimals. Beyond that, you hit the precision limit of 64-bit floating point, where extra digits become unreliable noise rather than real information.
What is the difference between √ and raising to the power of 1/2?
They are mathematically equivalent for non-negative numbers: √16 = 16^(1/2) = 4, and in general the nth root of x equals x^(1/n). The radical symbol (√) is the traditional notation, while fractional exponents come from algebra and are easier to use in formulas and programming. The one subtlety: some calculators treat (−8)^(1/3) differently from ∛(−8), so for negative bases the root notation is less ambiguous.
When would I use a cube root instead of a square root?
Cube roots (∛) reverse cubing (x³) and appear whenever you work with three-dimensional scaling. The most common example is volume: if a cube has a volume of 27 m³, its side length is ∛27 = 3 m. Cube roots also show up in finance (annualized growth over 3 years), fluid dynamics, and anywhere a quantity scales with the cube of a linear dimension. Square roots, by contrast, relate to areas and two-dimensional scaling.