Square Root Calculator

Mastering Square Roots and Radicals: Complete Guide

A square root of a number x is a value y such that y² = x. The principal square root is the non‑negative root, denoted by the radical symbol √. Every positive number has two square roots: a positive and a negative one. This Professional Square Root Calculator provides tools to compute square roots, nth roots, simplify radicals, and explore perfect squares—all with clear, step‑by‑step explanations.

Perfect Squares

A perfect square is an integer that is the square of an integer. The first few perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Recognizing perfect squares is essential for simplifying radicals and estimating square roots.

Nth Roots – Beyond the Square

The nth root of a number x is a value y such that yⁿ = x. Cube roots (n=3) and fourth roots are common. For even n, negative radicands produce imaginary results (involving i). Our Nth Root tab handles any integer root ≥2.

Properties of Radicals

• √(a × b) = √a × √b
• √(a / b) = √a / √b (b ≠ 0)
• √(a²) = |a|
• (√a)² = a (a ≥ 0)

Simplifying Radicals Step‑by‑Step

1. Find the prime factorization of the radicand.
2. Pair identical prime factors.
3. For each pair, move one factor outside the radical.
4. Multiply the factors outside and inside separately.