DIY Cost Tools
Square Root Calculator
MATH CALCULATOR
Calculate a number’s principal square root, cube root, and square-root classification.
How it works
The principal square root is the nonnegative number that produces the input when multiplied by itself. The cube root may be positive or negative, but this calculator limits the square-root result to real numbers.
Formulas
If r = √x, then r² = x.
Cube root c = ∛x, so c³ = x.
Example
√81 = 9 because 9 × 9 = 81. Since 9 is a whole number, 81 is a perfect square.
Important limits
- Negative inputs require complex numbers for a square root.
- Decimal results are rounded only for display.
- Very large inputs are limited by browser number precision.
Frequently asked questions
Why are there two square roots?
The equation x² = 81 has solutions ±9, but the principal square-root symbol √81 means the nonnegative value 9.
What is a perfect square?
It is an integer that equals another integer multiplied by itself.
Interpreting square roots
The calculator reports the principal, nonnegative square root for a nonnegative real input. A perfect square produces an integer result, while most other values produce an irrational decimal that must be rounded. Squaring the displayed result provides a useful check, allowing for the selected rounding precision.
Negative numbers do not have real square roots; they require complex numbers and the imaginary unit i. When simplifying radicals by hand, factor out perfect squares—for example, √72 = 6√2. Keep extra digits if the result feeds another calculation, especially in geometry, engineering, or statistics. Round only the final answer and match the number of significant figures to the accuracy of the original measurement.
Review notes
Results follow standard real-number root definitions. Last reviewed September 30, 2026.