What is a perfect square?

A perfect square is an integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Numbers like 2, 3, 5 are not perfect squares — their roots are irrational.

How do you simplify a radical?

Find the prime factorisation, then pull out pairs of prime factors. Each pair of identical primes under the radical becomes one factor outside. Example: √72 = √(4×9×2) = 2×3×√2 = 6√2.

The cube root ∛N is X such that X³=N. Example: ∛27=3. Cube roots exist for negative numbers: ∛(−8)=−2. Set the Root field to 3 in the calculator to compute cube roots.

Square Root Calculator — √ and Nth Root | Temporary Email Address generator edu email 2026

What is a Square Root?

The square root of a number N is a value X such that X × X = N. For example, √25 = 5 because 5² = 25. Every positive number has two square roots: a positive one (principal root) and a negative one. The square root symbol √ was first used in the 16th century and comes from the Latin word "radix" meaning root.

Our calculator handles not just square roots but any Nth root. The cube root (∛) of N finds X where X³ = N, and so on. It also simplifies radicals — for example √72 = 6√2 — and shows a step-by-step factor tree so you can understand the simplification process.

How to Use the Square Root Calculator

Enter a Number to find the root of (must be ≥ 0 for real results).
Set the Root value: 2 = square root, 3 = cube root, etc. Use the quick-select buttons.
Results appear instantly: decimal result, simplified radical form, inverse, and perfect-root status.
Read the Step-by-Step section to see how the radical is simplified via prime factorisation.

Why Use Our Square Root Calculator?

Simplified Radicals — Automatically reduces √72 to 6√2.
Any Nth Root — Not just square roots; supports cube, 4th, 5th, and higher roots.
Step-by-Step — Shows prime factorisation so you can learn the method.
Perfect Root Detection — Instantly tells you if the result is a whole number.
Nearby Perfect Squares — Puts your number in context on the number line.

Frequently Asked Questions

The square root of N is the number X such that X² = N. For example, √64 = 8 because 8 × 8 = 64. Every positive number has a positive principal square root. Negative numbers do not have real square roots — only complex ones (involving i = √−1). Zero has a square root of zero.

A perfect square is an integer whose square root is also an integer. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Numbers like 2, 3, 5, 7, 8 are not perfect squares — their square roots are irrational (non-terminating, non-repeating decimals). Recognising perfect squares helps simplify radicals in algebra and geometry.

To simplify √N: find the prime factorisation of N, then pair up prime factors. Each pair comes out from under the radical. For √72: 72 = 2×2×2×3×3. Group pairs: (2×2)×(3×3)×2. Each pair of 2s and 3s comes out: 2×3×√2 = 6√2. The number outside is the coefficient; the remaining unpaired primes stay under the radical.

The cube root of N (written ∛N) is X such that X³ = N. For example, ∛27 = 3 because 3³ = 27. Unlike square roots, cube roots exist for negative numbers: ∛−8 = −2. Cube roots appear in geometry (volume calculations), physics, and chemistry. Set the "Root" field to 3 in our calculator to compute cube roots.

Quick Facts

✓ Square, cube, and Nth roots
✓ Simplified radical form
✓ Step-by-step factor tree
✓ Perfect square detection
✓ Nearby perfect squares shown