Square Root & Cube Root Calculator

Square and cube roots, including non-perfect squares.

Overview

Computes square roots, cube roots, and nth roots of positive numbers with high floating-point precision and radical simplification.

Formula & Calculation Method

Square Root = sqrt(x) = x^(1/2) | Cube Root = cbrt(x) = x^(1/3) | Nth Root = x^(1/n)

Calculates the non-negative real root r such that r^n = x. Evaluates perfect squares and decimal approximations.

Worked Example: Finding the Square Root of 250

Inputs: Number x = 250 | Root Type = Square Root (n = 2)

  • Identify nearest perfect squares: 15² = 225 and 16² = 256
  • Estimate value between 15 and 16
  • High-precision Newton-Raphson calculation: sqrt(250) = 15.8113883
  • Radical simplification: sqrt(250) = sqrt(25 × 10) = 5 × sqrt(10)

Summary: sqrt(250) = 15.8114 | Simplified Radical = 5√10 | Perfect Square: No

Common Use Cases

  • Geometry & Trigonometry: Solve right triangle hypotenuses using the Pythagorean theorem (c = sqrt(a² + b²)).
  • Statistical Calculations: Compute standard deviation from sample variance.
  • Physics & Engineering: Evaluate kinetic energy, electrical RMS voltage, and wave frequency formulas.

Frequently Asked Questions

What is a perfect square?

An integer that is the square of another integer (e.g. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100).

How does the Newton-Raphson method calculate square roots?

It uses iterative approximations: x_(k+1) = 0.5 × (x_k + S / x_k), rapidly doubling correct decimal precision with each iteration.

Can a negative number have a real cube root?

Yes. For example, the cube root of -8 is -2, because (-2) × (-2) × (-2) = -8.

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