Square Root & Exponent Calculator
Perfect Squares Reference (1–20)
| n | n² | √(n²) = n |
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Square Root Calculator – √ and Exponent Guide
This free online square root calculator instantly finds the square root, cube root, Nth root, and Nth power of any number. Enter a number, optionally specify N for custom root or power calculations, and get all six results at once. A perfect squares reference table is included on the page for quick lookup.
What Is a Square Root?
The square root of a number x is the value that, when multiplied by itself, gives x. Written as √x, it is the inverse operation of squaring. For example, √144 = 12 because 12 × 12 = 144. Every positive number has two square roots — a positive and a negative — but by convention, the principal (positive) square root is what calculators return.
Square roots of negative numbers are not real numbers. In the complex number system they involve the imaginary unit i, but this calculator works in the real number domain only.
Cube Roots and Nth Roots
The cube root (∛x) is the number that, multiplied by itself three times, equals x. Unlike square roots, cube roots are defined for negative numbers: ∛(−8) = −2. The general Nth root of x equals x^(1/N). This calculator supports any positive integer N you enter.
Perfect Squares
A perfect square is any integer that is the square of another integer. The first ten are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Recognizing perfect squares speeds up mental arithmetic and is useful in simplifying radical expressions. A full reference table for 1 through 20 is included in the tool above.
Applications of Square Roots
- Geometry: Finding the side length of a square from its area (side = √area).
- Physics: Velocity calculations in kinetic energy formulas (v = √(2E/m)).
- Statistics: Standard deviation involves a square root.
- Engineering: RMS (root mean square) voltage and signal processing.
- Finance: Compound interest period calculations use Nth roots.