Square Root Calculator

How the Square Root Calculator Works

This Square Root Calculator focuses on clear, exact results for a single number at a time. It validates your input, applies the principal square root for non-negative values, simplifies radicals using prime factorization, and presents both exact and decimal forms side by side so you can see the full picture at a glance.

Step 1: Input Validation & Parsing

When you enter a value for x, the calculator first checks that it is a finite numeric input. Any non-numeric, empty, or infinite values are rejected with a clear error message instead of returning misleading results. This makes the tool safe to use for homework, teaching, and quick checks.

Step 2: Handling Zero, Positive, and Negative Values

The behavior changes depending on the sign of x:

• x = 0: returns an exact result of 0 and a decimal value formatted to your chosen precision.
• x > 0: computes the principal (non-negative) square root, then looks for perfect-square factors so √x can be written in simplified radical form a√b. If b = 1, the calculator labels x as a perfect square and shows the whole-number root.
• x < 0: explains that no real square root exists and expresses the result as an imaginary number. It simplifies √|x| into a√b and then presents the complex root as ai√b, along with the decimal magnitude for reference.

Step 3: Simplifying Radicals with Prime Factorization

For positive integers, the calculator uses prime factorization to simplify radicals:

• Factors x into its prime components.
• Groups primes into pairs; each full pair moves outside the radical as part of a.
• Any unpaired primes remain inside the radical as b, ensuring b has no perfect-square factors.

This process is reflected in the explanatory panel so you can see exactly why a√b is correct instead of treating the simplification as a black box.

Exact Forms vs Decimal Approximations

The calculator keeps exact and approximate values clearly separated. You can choose a decimal precision from 0 to 12 places for the numeric approximation, while the simplified radical or imaginary form remains exact. This is especially useful for exams, proofs, and situations where rounding errors matter.

Scope, Assumptions & Best Uses

• Uses the principal square root convention for real numbers (no ± pair for basic output).
• Treats radical simplification on integer inputs using standard algebra rules.
• Provides imaginary outputs for negative inputs in a concise a·i√b form.
• Designed as a fast checker and learning aid, not a full symbolic algebra system.

Use this Square Root Calculator to verify homework, test if values are perfect squares, simplify radicals into a√b form, and understand how imaginary roots arise from negative inputs—all in a single, SEO-friendly page.